<h1>A new wrapped exponential distribution</h1>
			<ul class="item-list">
	        	        <li>
	            Abdullah Yilmaz	            	            	            	            <sup aria-label="Affiliated with Department of Actuarial Sciences, Kirikkale University, Kirikkale, 71450, TR">
	                1	            </sup>
	            	        </li>
	        	        <li>
	            Cenker Biçer	            	            <abbr title="This is the corresponding author for this article">*</abbr>
	            	            	                <a href="mailto:cbicer@kku.edu.tr" class="tiny-icon email-link mx-1" title="Email Cenker Biçer">
	                    Email
	                </a>
	            	            	            <sup aria-label="Affiliated with Department of Statistics, Kirikkale University, Kirikkale, 71450, TR">
	                2	            </sup>
	            	        </li>
	        	    </ul>
	    	    <ol class="affiliations" aria-hidden="true">
	        <li>Department of Actuarial Sciences, Kirikkale University, Kirikkale, 71450, TR</li><li>Department of Statistics, Kirikkale University, Kirikkale, 71450, TR</li>	    </ol>
	    
<h2>Abstract</h2>
<p>We introduce a new wrapped exponential distribution named transmuted wrapped exponential (TWE) distribution, for the modeling of circular datasets by using the Transmutation Rank-Map method. This method is employed for the first time for a wrapped distribution with this study. The introduced distribution is more flexible than traditional wrapped exponential distribution. The paper provides the explicit form of important distributional properties of the introduced distribution such as expectation, median, moments, characteristic function, quantile function, hazard rate function and stress-strength reliability. Rényi and Shannon entropies are also obtained. The statistical inference problem for the TWE distribution is investigated using maximum likelihood, least squares and weighted least squares and comparative numerical study results are presented. Furthermore, we present a real dataset analysis.</p><hr/><section><h2>Introduction</h2>
<p>In statistical meaning, it is known that the performance of a statistical analysis depends on the selected model distribution for a data set. If the selected distribution is an optimal model to data, then the obtained statistical inference from the dataset is the best. Because of this, a number of researchers suggested adding extra parameters to the distributions in order to be able to create more flexible distributions. Quadratic rank transmutation map (QRTM) technique is one of these methods. Depending on a base distribution, the transmuted distribution is obtained as follows.</p>
<p>Suppose that 
<em>X</em>
 is a real-valued random variable and also 
<span id="IEq1"><mml:math id="IEq1_Math"><mml:mrow><mml:mi>G</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq1_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 and 
<span id="IEq2"><mml:math id="IEq2_Math"><mml:mrow><mml:mi>g</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq2_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 are the cumulative distribution function (cdf) and the probability density function (pdf) of 
<em>X</em>
, respectively. Then
<section id="Equ1"><mml:math display="block" id="Equ1_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>F</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mi>G</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>G</mml:mi><mml:msup><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ1_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} F\left( x\right) =\left( 1+\Lambda \right) G\left( x\right) -\Lambda G\left( x\right) ^{2} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ1.gif"/></section>
and
<section id="Equ2"><mml:math display="block" id="Equ2_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>f</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mfenced close="]" open="[" separators=""><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>G</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width="4pt"/><mml:mspace width="4pt"/><mml:mspace width="4pt"/><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ2_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} f\left( x\right) =g\left( x\right) \left[ \left( 1+\Lambda \right) -2\Lambda G\left( x\right) \right] ,\ \ \ -1\le \Lambda \le 1 \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ2.gif"/></section>
are called a transmuted cdf and pdf, respectively, depending on base cdf 
<span id="IEq3"><mml:math id="IEq3_Math"><mml:mrow><mml:mi>G</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq3_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 and pdf 
<span id="IEq4"><mml:math id="IEq4_Math"><mml:mrow><mml:mi>g</mml:mi><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq4_TeX"><![CDATA[\documentclass[12pt]{minimal}
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, where 
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq5.gif"/></span>
 is the transmuting parameter [
<a href="#CR18"><sup>18</sup></a>
]. So far, it has been shown by the conducted studies that the QRTM distributions obtained from the base distributions are better models to the dataset than the base distributions, because QRTM distributions have more parameters and they are more flexible than the base distribution. Khan et al. [
<a href="#CR10"><sup>10</sup></a>
] proposed the Transmuted Generalized Exponential distribution using the QRTM method, and they compared their model with existing lifetime distributions such as, TGE, exponentiated Weibull (EW), modified Weibull (MW), generalized exponential (GE), weighted exponential, extended exponential (EE), Weibull (W), and Power generalized Weibull (PGW). Kemaloglu and Yilmaz [
<a href="#CR8"><sup>8</sup></a>
] presented the Transmuted two-parameter Lindley distribution (TTLD) as a new lifetime distribution. They studied some important statistical properties of the TTLD. Aryal and Tsokos [
<a href="#CR2"><sup>2</sup></a>
] introduced the transmuted Weibull distribution and studied its mathematical properties. In 2013, Merovci applied the QRTM to the exponentiated exponential distribution and introduced the transmuted exponentiated exponential distribution as a lifetime distribution [
<a href="#CR13"><sup>13</sup></a>
]. We refer the interested reader to [
<a href="#CR3"><sup>3</sup></a>
, 
<a href="#CR4"><sup>4</sup></a>
, 
<a href="#CR9"><sup>9</sup></a>
, 
<a href="#CR14"><sup>14</sup></a>
, 
<a href="#CR16"><sup>16</sup></a>
, 
<a href="#CR17"><sup>17</sup></a>
, 
<a href="#CR20"><sup>20</sup></a>
] and the references therein for more literature information on the transmuted families of distributions.</p>
<p>The main goal of this study is to create a more flexible distribution called transmuted wrapped exponential (TWE) for the modeling of circular data based on QRTM method. The QRTM technique is employed for the first time for a wrapped distribution with this study.</p>
<p>The rest of this paper is organized as follows: In "
<a href="#Sec2"><sup>TWE distribution</sup></a>
" section, the cdf and pdf of TWE distribution are obtained. In addition, some important properties of the TWE distribution such as trigonometric moments, characteristic function, location, dispersion, median, skewness, kurtosis, modality behavior, order statistics, entropy, stress-strength reliability and hazard rate function are studied in that section. The statistical inference problem for the TWE distribution according to the maximum likelihood (ML), the least squares (LS) and the weighted least squares (WLS) method are discussed in "
<a href="#Sec12"><sup>Inference</sup></a>
" section. A series of simulation experiments for comparing the performance of the obtained estimators are performed in "
<a href="#Sec15"><sup>Monte Carlo simulation study</sup></a>
" section. We analyze a real-life dataset from the literature for illustrative purposes in "
<a href="#Sec16"><sup>Application to real data</sup></a>
" section. Finally, the last section of the paper concludes the study.</p></section>
<section><h2>TWE distribution</h2>
<p>The wrapping method is a well-known approach to obtain a circular distribution based on a distribution family. The wrapped distributions play quite an important role in the modeling of circular data. Jammalamadaka and Kozubowski [
<a href="#CR5"><sup>5</sup></a>
] introduced the wrapped exponential (WE) distribution with following pdf and cdf,
<section id="Equ3"><mml:math display="block" id="Equ3_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:msub><mml:mi>X</mml:mi><mml:mi>w</mml:mi></mml:msub></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ3_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} f_{X_{w}}\left( \theta \right) =\frac{\lambda {\mathrm{e}}^{-\lambda \theta }}{1-{\mathrm{e}}^{-2\pi \lambda }} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ3.gif"/></section>
and
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				\begin{document}$$\begin{aligned} F_{X_{w}}\left( \theta \right) =\frac{1-{\mathrm{e}}^{-\lambda \theta }}{1-{\mathrm{e}}^{-2\pi \lambda }}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ4.gif"/></section>
respectively, where 
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				\begin{document}$$\lambda >0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq6.gif"/></span>
 and 
<span id="IEq7"><mml:math id="IEq7_Math"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mfenced close=")" open="[" separators=""><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq7_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\theta \in \left[ 0,2\pi \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq7.gif"/></span>
. The main motivation of this study is to obtain a more flexible circular distribution than WE to the optimal modeling of circular data. Therefore, by using formulas (
<a href="#Equ3"><sup>3</sup></a>
) and (
<a href="#Equ4"><sup>4</sup></a>
) in QRTM method, we obtain cdf and pdf of a TWE distributed random variable 
<span id="IEq8"><mml:math id="IEq8_Math"><mml:mi mathvariant="normal">Θ</mml:mi></mml:math><tex-math id="IEq8_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq8.gif"/></span>
 as
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				\begin{document}$$\begin{aligned} F_{\Theta }\left( \theta \right) =\frac{\left( {\mathrm{e}}^{-\lambda \theta }-1\right) \left( c+\Lambda \left( 1+c-{\mathrm{e}}^{-\theta \lambda }\right) \right) }{c^{2}} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ5.gif"/></section>
and
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				\begin{document}$$\begin{aligned} f_{\Theta }\left( \theta \right) =\frac{2\lambda \Lambda {\mathrm{e}}^{-\theta \lambda }\left( {\mathrm{e}}^{-\theta \lambda }-1\right) }{c^{2}}-\frac{\lambda {\mathrm{e}}^{-\theta \lambda }\left( \Lambda +1\right) }{c}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ6.gif"/></section>
respectively, where and through the paper 
<span id="IEq9"><mml:math id="IEq9_Math"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq9_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$c={\mathrm{e}}^{-2\pi \lambda }-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq9.gif"/></span>
, 
<span id="IEq10"><mml:math id="IEq10_Math"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq10_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda >0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq10.gif"/></span>
, 
<span id="IEq11"><mml:math id="IEq11_Math"><mml:mrow><mml:mfenced close="|" open="|"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq11_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\left| \Lambda \right| \le 1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq11.gif"/></span>
 and 
<span id="IEq12"><mml:math id="IEq12_Math"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mfenced close=")" open="[" separators=""><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq12_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\theta \in \left[ 0,2\pi \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq12.gif"/></span>
. From now on, a random variable 
<span id="IEq13"><mml:math id="IEq13_Math"><mml:mi mathvariant="normal">Θ</mml:mi></mml:math><tex-math id="IEq13_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq13.gif"/></span>
 distributed the TWE with parameters 
<span id="IEq14"><mml:math id="IEq14_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq14_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq14.gif"/></span>
 and 
<span id="IEq15"><mml:math id="IEq15_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq15_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq15.gif"/></span>
 will be indicated as 
<span id="IEq16"><mml:math id="IEq16_Math"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq16_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta \sim {\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq16.gif"/></span>
. Figure 
<a href="#Fig1"><sup>1</sup></a>
 illustrates the some of the possible shapes of the pdf of a TWE distribution for different values of the parameters 
<span id="IEq17"><mml:math id="IEq17_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq17_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq17.gif"/></span>
 and 
<span id="IEq18"><mml:math id="IEq18_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq18_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq18.gif"/></span>
.
<figure id="Fig1"><h3>Fig. 1</h3>
<figcaption><p>Pdf of transmuted wrapped exponential distribution for different values of 
<span id="IEq19"><mml:math id="IEq19_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq19_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq19.gif"/></span>
 and 
<span id="IEq20"><mml:math id="IEq20_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq20_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq20.gif"/></span></p></figcaption>
<img src="40096_2018_268_Fig1_HTML.png" /></figure></p>
<p>As it can be seen from Fig. 
<a href="#Fig1"><sup>1</sup></a>
, the TWE distribution is a unimodal distribution. When 
<span id="IEq21"><mml:math id="IEq21_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq21_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda >0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq21.gif"/></span>
, the mode of the distribution is zero; otherwise, it differs from zero for some values of the 
<span id="IEq22"><mml:math id="IEq22_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq22_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq22.gif"/></span>
, see "
<a href="#Sec6"><sup>Modality Behavior</sup></a>
" section. We can say that the distribution has got often the negative skewness (we say anticlockwise skewness). The parameter 
<span id="IEq23"><mml:math id="IEq23_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq23_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq23.gif"/></span>
 plays an important role in the mean and variance of the TWE distribution as a heritage of its task in the exponential distribution.</p>
<section><h2>Characteristic function</h2>
<p>The characteristic function of 
<span id="IEq24"><mml:math id="IEq24_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq24_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq24.gif"/></span>
 distribution is
<section id="Equ7"><mml:math display="block" id="Equ7_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>φ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>p</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ7_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \varphi _{\Theta }(p)&=\varphi _{p}=E({\mathrm{e}}^{ip\Theta }) \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ7.gif"/></section>
<section id="Equ8"><mml:math display="block" id="Equ8_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mfenced close="]" open="[" separators=""><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>λ</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi>λ</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mfenced><mml:mfenced close="]" open="[" separators=""><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ8_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned}&=\frac{\lambda \left( \lambda +ip\right) \left( 2\Lambda +c+c\Lambda \right) \left[ \left( c+1\right) {\mathrm{e}}^{2i\pi p}-1\right] }{c^{2}\left( \lambda ^{2}+p^{2}\right) }\nonumber \\&\quad -\frac{2\lambda \Lambda \left( 2\lambda +ip\right) \left[ \left( c+1\right) ^{2}{\mathrm{e}}^{2i\pi p}-1\right] }{c^{2}\left( 4\lambda ^{2} +p^{2}\right) } \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ8.gif"/></section>
However, since a circular random variable is periodic, 
<span id="IEq25"><mml:math id="IEq25_Math"><mml:mi mathvariant="normal">Θ</mml:mi></mml:math><tex-math id="IEq25_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq25.gif"/></span>
 and 
<span id="IEq26"><mml:math id="IEq26_Math"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:math><tex-math id="IEq26_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta +2\pi $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq26.gif"/></span>
 have the same distribution, and 
<em>p</em>
 must be restricted to the integer values [
<a href="#CR12"><sup>12</sup></a>
].</p></section>
<section><h2>Trigonometric moments</h2>
<p>The value of the characteristic function of the circular random variable 
<span id="IEq27"><mml:math id="IEq27_Math"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq27_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta \sim {\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq27.gif"/></span>
 at an integer 
<em>p</em>
 is called the 
<em>p</em>
th trigonometric moment. One can also write 
<em>p</em>
th trigonometric moments in terms of 
<span id="IEq28"><mml:math id="IEq28_Math"><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><tex-math id="IEq28_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\alpha _{p}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq28.gif"/></span>
 and 
<span id="IEq29"><mml:math id="IEq29_Math"><mml:msub><mml:mi>β</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><tex-math id="IEq29_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\beta _{p}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq29.gif"/></span>
<section id="Equ22"><mml:math display="block" id="Equ22_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>φ</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>β</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="4pt"/><mml:mspace width="4pt"/><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mspace width="0.166667em"/><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ22_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \varphi _{p}=\varphi _{\Theta }(p)=\alpha _{p}+i\beta _{p},\ \ p=0,\pm \, 1,\pm \, 2,\ldots . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ22.gif"/></section>
where 
<span id="IEq30"><mml:math id="IEq30_Math"><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><tex-math id="IEq30_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\alpha _{p}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq30.gif"/></span>
 is 
<em>p</em>
th cosine moment defined as 
<span id="IEq31"><mml:math id="IEq31_Math"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>cos</mml:mo><mml:mi>p</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq31_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\alpha _{p}=E(\cos p\Theta )$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq31.gif"/></span>
 and 
<span id="IEq32"><mml:math id="IEq32_Math"><mml:msub><mml:mi>β</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><tex-math id="IEq32_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\beta _{p}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq32.gif"/></span>
 is 
<em>p</em>
th sine moment defined as 
<span id="IEq33"><mml:math id="IEq33_Math"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>sin</mml:mo><mml:mi>p</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq33_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\beta _{p}=E(\sin p\Theta )$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq33.gif"/></span>
. Hence, the 
<em>p</em>
th cosine moment of 
<span id="IEq34"><mml:math id="IEq34_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq34_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq34.gif"/></span>
 distribution is
<section id="Equ9"><mml:math display="block" id="Equ9_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>6</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ9_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \alpha _{p}=\frac{\lambda ^{2}\left( 4c\lambda ^{2}-6p^{2}\Lambda +cp^{2} -3cp^{2}\Lambda \right) }{c\left( 4\lambda ^{2}+p^{2}\right) \left( \lambda ^{2}+p^{2}\right) } \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ9.gif"/></section>
and 
<em>p</em>
th sine moment is
<section id="Equ10"><mml:math display="block" id="Equ10_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mi>p</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ10_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \beta _{p}=\frac{\lambda p\left( 4\lambda ^{2}\Lambda +4c\lambda ^{2}-2\Lambda p^{2}+cp^{2}+2c\lambda ^{2}\Lambda -cp^{2}\Lambda \right) }{c\left( 4\lambda ^{2}+p^{2}\right) \left( \lambda ^{2}+p^{2}\right) }, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ10.gif"/></section>
where 
<span id="IEq35"><mml:math id="IEq35_Math"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mspace width="0.166667em"/><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>±</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:mrow></mml:math><tex-math id="IEq35_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$p=0,\pm \, 1,\pm \, 2,\ldots $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq35.gif"/></span>
.</p></section>
<section><h2>Location, dispersion and median</h2>
<p>The 
<em>p</em>
th trigonometric moment of 
<span id="IEq36"><mml:math id="IEq36_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq36_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq36.gif"/></span>
 can be expressed in 
<span id="IEq37"><mml:math id="IEq37_Math"><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mi>μ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math id="IEq37_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\varphi _{p}=\rho _{p}{\mathrm{e}}^{i\mu _{p}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq37.gif"/></span>
 where 
<span id="IEq38"><mml:math id="IEq38_Math"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mtext>atan</mml:mtext><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>α</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:msubsup><mml:mi>β</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq38_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\mu _{p} ={\text {atan}}\left( \alpha _{p}\beta _{p}^{-1}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq38.gif"/></span>
 and 
<span id="IEq39"><mml:math id="IEq39_Math"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>β</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:msqrt></mml:mrow></mml:math><tex-math id="IEq39_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\rho _{p}=\sqrt{\alpha _{p}^{2}+\beta _{p}^{2}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq39.gif"/></span>
. 
<span id="IEq40"><mml:math id="IEq40_Math"><mml:msub><mml:mi>φ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><tex-math id="IEq40_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\varphi _{p}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq40.gif"/></span>
 has a special meaning for 
<span id="IEq41"><mml:math id="IEq41_Math"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq41_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$p=1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq41.gif"/></span>
. The 
<span id="IEq42"><mml:math id="IEq42_Math"><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math><tex-math id="IEq42_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\rho _{1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq42.gif"/></span>
 and the 
<span id="IEq43"><mml:math id="IEq43_Math"><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math><tex-math id="IEq43_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\mu _{1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq43.gif"/></span>
 are called angular concentration and mean direction, respectively. Here 
<span id="IEq44"><mml:math id="IEq44_Math"><mml:mrow><mml:mtext>atan</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mo>.</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><tex-math id="IEq44_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\text {atan}}( .) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq44.gif"/></span>
 is quadrant inverse tangent function and defined as
<section id="Equ23"><mml:math display="block" id="Equ23_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mtext>atan</mml:mtext><mml:mfenced close=")" open="(" separators=""><mml:mi>y</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close="" open="{" separators=""><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:msup><mml:mo>tan</mml:mo><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>x</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>y</mml:mi></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mspace width="1em"/><mml:mi>y</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mi>π</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mspace width="1em"/><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:msup><mml:mo>tan</mml:mo><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>x</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>y</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>π</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mspace width="1em"/><mml:mi>y</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:msup><mml:mo>tan</mml:mo><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>x</mml:mi><mml:mo stretchy="false">/</mml:mo><mml:mi>y</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mspace width="1em"/><mml:mi>y</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mtext>undefined</mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow/><mml:mspace width="1em"/><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ23_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} {\text {atan}}\left( y/x\right) =\left\{ \begin{array} [c]{ll} \tan ^{-1}\left( x/y\right) , &{}\quad y>0,x\ge 0\\ \pi /2, &{}\quad y=0,x>0\\ \tan ^{-1}\left( x/y\right) +\pi , &{}\quad y<0\\ \tan ^{-1}\left( x/y\right) +2\pi , &{}\quad y\ge 0,x<0\\ {\text {undefined}}, &{}\quad y=0,x=0 \end{array} \right. . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ23.gif"/></section>
Mean direction of 
<span id="IEq45"><mml:math id="IEq45_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq45_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq45.gif"/></span>
 distribution is
<section id="Equ11"><mml:math display="block" id="Equ11_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mtext>atan</mml:mtext><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:mi>c</mml:mi><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>6</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ11_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \mu _{1}={\text {atan}}\left( \frac{c-2\Lambda +4\lambda ^{2} \Lambda +4c\lambda ^{2}-c\Lambda +2c\lambda ^{2}\Lambda }{\lambda \left( 4c\lambda ^{2}-6\Lambda +c-3c\Lambda \right) }\right) . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ11.gif"/></section>
The mean direction vector gives information about the mean of the distribution as an analogy of the mean in the linear models. The length of this vector is a measure of dispersion around the mean and corresponds to the usual standard deviation or variance in linear models. The angular concentration for 
<span id="IEq46"><mml:math id="IEq46_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq46_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq46.gif"/></span>
 distribution is
<section id="Equ12"><mml:math display="block" id="Equ12_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:msqrt></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ12_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \rho _{1}=\sqrt{\frac{\lambda ^{2}\left( 2\Lambda -c+c\Lambda \right) ^{2}+4c^{2}\lambda ^{4}}{c^{2}\left( 4\lambda ^{2}+1\right) \left( \lambda ^{2}+1\right) }.} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ12.gif"/></section>
For a circular model, the circular variance is calculated as 
<span id="IEq47"><mml:math id="IEq47_Math"><mml:mrow><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><tex-math id="IEq47_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$V=1-\rho _{1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq47.gif"/></span>
. Hence, using the (
<a href="#Equ12"><sup>12</sup></a>
), the circular variance of 
<span id="IEq48"><mml:math id="IEq48_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq48_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq48.gif"/></span>
 distribution is obtained as
<section id="Equ13"><mml:math display="block" id="Equ13_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:msqrt></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ13_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} V=1-\sqrt{\frac{\lambda ^{2}\left( 2\Lambda -c+c\Lambda \right) ^{2} +4c^{2}\lambda ^{4}}{c^{2}\left( 4\lambda ^{2}+1\right) \left( \lambda ^{2}+1\right) }.} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ13.gif"/></section>
Also, the circular standard deviation calculated as 
<span id="IEq49"><mml:math id="IEq49_Math"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>ln</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msqrt></mml:mrow></mml:math><tex-math id="IEq49_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\sigma =\sqrt{-2\ln \rho _{1}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq49.gif"/></span>
 and calculated for 
<span id="IEq50"><mml:math id="IEq50_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq50_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq50.gif"/></span>
 distribution as
<section id="Equ24"><mml:math display="block" id="Equ24_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mfenced close="]" open="[" separators=""><mml:mo>-</mml:mo><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac></mml:mfenced></mml:mfenced><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ24_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \sigma =\left[ -\ln \left( \frac{\lambda ^{2}\left( 2\Lambda -c+c\Lambda \right) ^{2}+4c^{2}\lambda ^{4}}{c^{2}\left( 4\lambda ^{2}+1\right) \left( \lambda ^{2}+1\right) }\right) \right] ^{\frac{1}{2}}. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ24.gif"/></section>
The quantile function of 
<span id="IEq51"><mml:math id="IEq51_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq51_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq51.gif"/></span>
 can be easily obtained from the solution of equation 
<span id="IEq52"><mml:math id="IEq52_Math"><mml:mrow><mml:mi>F</mml:mi><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq52_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$F\left( \theta \right) -u=0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq52.gif"/></span>
 with respect to 
<span id="IEq53"><mml:math id="IEq53_Math"><mml:mi>θ</mml:mi></mml:math><tex-math id="IEq53_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\theta $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq53.gif"/></span>
 as
<section id="Equ14"><mml:math display="block" id="Equ14_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>Q</mml:mi><mml:mfenced close=")" open="("><mml:mi>u</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>λ</mml:mi></mml:mfrac><mml:mo>ln</mml:mo><mml:mfenced close="]" open="[" separators=""><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ14_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} Q\left( u\right) =-\frac{1}{\lambda }\ln \left[ \frac{2\Lambda +c-c\sqrt{2\Lambda +\Lambda ^{2}-4\Lambda u+1}+c\Lambda }{2\Lambda }\right] , \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ14.gif"/></section>
where 
<span id="IEq54"><mml:math id="IEq54_Math"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq54_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$u\in \left( 0,1\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq54.gif"/></span>
. Then the median direction of a circular distribution is a value 
<em>M</em>
 such that 
<span id="IEq55"><mml:math id="IEq55_Math"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:msubsup><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="normal">Θ</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq55_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\int _{0}^{M}f_{\Theta }\left( \theta \right) {\mathrm{d}}\theta =\int _{M}^{2\pi }f_{\Theta }\left( \theta \right) {\mathrm{d}}\theta =0.5$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq55.gif"/></span>
. The median of 
<span id="IEq56"><mml:math id="IEq56_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq56_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq56.gif"/></span>
 distribution is obtained from equation 
<span id="IEq57"><mml:math id="IEq57_Math"><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mi>Q</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>0.5</mml:mn></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq57_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$M=Q\left( 0.5\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq57.gif"/></span>
 as
<section id="Equ15"><mml:math display="block" id="Equ15_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>λ</mml:mi></mml:mfrac><mml:mo>ln</mml:mo><mml:mfenced close="]" open="[" separators=""><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mfenced></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ15_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} M=-\frac{1}{\lambda }\ln \left[ \frac{c\left( \Lambda +1-\sqrt{\Lambda ^{2} +1}\right) }{2\Lambda }+1\right] . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ15.gif"/></section></p></section>
<section><h2>Modality behavior</h2>
<p><span id="IEq58"><mml:math id="IEq58_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq58_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq58.gif"/></span>
 is a unimodal distribution for 
<span id="IEq59"><mml:math id="IEq59_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq59_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda \ne 0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq59.gif"/></span>
. The critical value of its pdf (
<a href="#Equ6"><sup>6</sup></a>
) can be immediately calculated as
<section id="Equ25"><mml:math display="block" id="Equ25_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>λ</mml:mi></mml:mfrac><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>∈</mml:mo><mml:mfenced close=")" open="[" separators=""><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ25_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \theta _{0}=-\frac{1}{\lambda }\ln \left( \frac{2\Lambda +c+\Lambda c}{4\Lambda }\right) \in \left[ 0,2\pi \right) . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ25.gif"/></section>
On the other hand, for
<section id="Equ26"><mml:math display="block" id="Equ26_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow><mml:mo>″</mml:mo></mml:msubsup><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>θ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ26_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} f_{\Theta }^{\prime \prime }\left( \theta _{0}\right) =\frac{\lambda ^{3}\left( 2\Lambda +c+\Lambda c\right) ^{2}}{4\Lambda c^{2}}<0 \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ26.gif"/></section>
the parameter 
<span id="IEq60"><mml:math id="IEq60_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq60_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq60.gif"/></span>
 must be negative. Thus, the mode of 
<span id="IEq61"><mml:math id="IEq61_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq61_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq61.gif"/></span>
, say 
<span id="IEq62"><mml:math id="IEq62_Math"><mml:msub><mml:mi>θ</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math><tex-math id="IEq62_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\theta _{T}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq62.gif"/></span>
, is
<section id="Equ27"><mml:math display="block" id="Equ27_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>λ</mml:mi></mml:mfrac><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ27_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \theta _{T}=-\frac{1}{\lambda }\ln \left( \frac{2\Lambda +c+\Lambda c}{4\Lambda }\right) \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ27.gif"/></section>
when 
<span id="IEq63"><mml:math id="IEq63_Math"><mml:mrow><mml:mfrac><mml:mi>c</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>&lt;</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:mi>c</mml:mi></mml:mfenced></mml:mfrac></mml:mrow></mml:math><tex-math id="IEq63_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\frac{c}{3c+2}<\Lambda <\frac{c}{\left( 2-c\right) }$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq63.gif"/></span>
 and 0 otherwise.</p></section>
<section><h2>Skewness and kurtosis</h2>
<p>For a circular model, the 
<em>p</em>
th central cosine and sine moments are
<span id="IEq64"><mml:math id="IEq64_Math"><mml:mrow><mml:mspace width="4pt"/><mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mfenced close="]" open="[" separators=""><mml:mo>cos</mml:mo><mml:mi>p</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>θ</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq64_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\ {\bar{\alpha }}_{p}=E\left[ \cos p\left( \theta -\mu _{1}\right) \right] $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq64.gif"/></span>
 and 
<span id="IEq65"><mml:math id="IEq65_Math"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mfenced close="]" open="[" separators=""><mml:mo>sin</mml:mo><mml:mi>p</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>θ</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq65_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\bar{\beta }}_{p}=E\left[ \sin p\left( \theta -\mu _{1}\right) \right] $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq65.gif"/></span>
, respectively [
<a href="#CR12"><sup>12</sup></a>
]. As a measure of asymmetry, skewness coefficient is calculated by 
<span id="IEq66"><mml:math id="IEq66_Math"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">γ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math id="IEq66_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\varvec{\gamma }}_{1}={\bar{\beta }}_{2}V^{-3/2}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq66.gif"/></span>
 for a circular distribution. Hence, the skewness coefficient of 
<span id="IEq67"><mml:math id="IEq67_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq67_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq67.gif"/></span>
 is obtained as
<section id="Equ16"><mml:math display="block" id="Equ16_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">γ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mfenced close="" open="[" separators=""><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>sin</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>cos</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mfenced></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mfenced close="]" open="" separators=""><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mo>cos</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mo>sin</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ16_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} {\varvec{\gamma }}_{1}&=-V^{-3/2}\left[ \frac{\lambda \left( \lambda \sin 2\mu _{1}-2\cos 2\mu _{1}\right) \left( 2\Lambda +c+c\Lambda \right) }{c\left( \lambda ^{2}+4\right) }\right. \\&\quad \left. -\frac{\lambda \Lambda \left( c+2\right) \left( \cos 2\mu _{1}-\lambda \sin 2\mu _{1}\right) }{c\left( \lambda ^{2}+1\right) }\right] .\nonumber \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ16.gif"/></section>
Kurtosis of a circular distribution is 
<span id="IEq68"><mml:math id="IEq68_Math"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">γ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mover accent="true"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">¯</mml:mo></mml:mrow></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msubsup><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mn>4</mml:mn></mml:msubsup></mml:mfenced><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math id="IEq68_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\varvec{\gamma }}_{2}=\left( {\bar{\alpha }}_{2}-\rho _{1}^{4}\right) \left( 1-\rho _{1}\right) ^{-2}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq68.gif"/></span>
. Therefore, kurtosis coefficient of 
<span id="IEq69"><mml:math id="IEq69_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq69_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq69.gif"/></span>
 is obtained as
<section id="Equ17"><mml:math display="block" id="Equ17_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">γ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>V</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mfenced close="" open="[" separators=""><mml:mfrac><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mi>λ</mml:mi><mml:mo>sin</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>+</mml:mo><mml:mspace width="0.166667em"/><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mi>λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mn>6</mml:mn><mml:mi>λ</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mi>λ</mml:mi><mml:mo>cos</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mfenced close="]" open="" separators=""><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mfrac><mml:msup><mml:mfenced close="]" open="[" separators=""><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>4</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ17_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} {\varvec{\gamma }}_{2}&=V^{-2}\left[ \frac{\lambda ^{2}\left( 2\Lambda +2c+\Lambda c\right) +2c\left( 1-\Lambda \right) -4\Lambda }{c\left( \lambda ^{4}+5\lambda ^{2}+4\right) }\lambda \sin \left( 2\mu _{1}\right) \right. \nonumber \\&\quad +\,\frac{c\lambda \left( 1+\lambda ^{2}-3\Lambda \right) -6\lambda \Lambda }{c\left( \lambda ^{4}+5\lambda ^{2}+4\right) }\lambda \cos \left( 2\mu _{1}\right) \nonumber \\&\quad \left. -\,\frac{\left[ \left( 2\Lambda -c+\Lambda c\right) ^{2}+4\lambda ^{4}c^{2}\right] ^{2}}{c^{4}\left( 4\lambda ^{2}+1\right) ^{2}\left( \lambda ^{2}+1\right) ^{2}}\right] . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ17.gif"/></section>
Figure 
<a href="#Fig2"><sup>2</sup></a>
 represents the contour plots of circular variance (
<em>V</em>
) , skewness 
<span id="IEq70"><mml:math id="IEq70_Math"><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>γ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:math><tex-math id="IEq70_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\left( \gamma _{1}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq70.gif"/></span>
 and kurtosis 
<span id="IEq71"><mml:math id="IEq71_Math"><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:math><tex-math id="IEq71_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\left( \gamma _{2}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq71.gif"/></span>
 of TWE distribution.
<figure id="Fig2"><h3>Fig. 2</h3>
<figcaption><p>Contour plots of circular variance (
<em>V</em>
) , skewness 
<span id="IEq72"><mml:math id="IEq72_Math"><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>γ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:math><tex-math id="IEq72_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\left( \gamma _1\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq72.gif"/></span>
 and kurtosis 
<span id="IEq73"><mml:math id="IEq73_Math"><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:math><tex-math id="IEq73_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\left( \gamma _2\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq73.gif"/></span>
 coefficient of TWE distribution</p></figcaption>
<img src="40096_2018_268_Fig2_HTML.png" /></figure></p>
<p>In general, for a constant value of 
<span id="IEq74"><mml:math id="IEq74_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq74_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq74.gif"/></span>
, it can be seen from Fig. 
<a href="#Fig2"><sup>2</sup></a>
 that when the 
<span id="IEq75"><mml:math id="IEq75_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq75_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq75.gif"/></span>
 increases, the circular variance decreases. However, this is not true for some negative values of 
<span id="IEq76"><mml:math id="IEq76_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq76_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq76.gif"/></span>
. Similarly, for a constant value of 
<span id="IEq77"><mml:math id="IEq77_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq77_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq77.gif"/></span>
, when 
<span id="IEq78"><mml:math id="IEq78_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq78_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq78.gif"/></span>
 increases, the circular variance decreases. As in the circular variance, the skewness decreases when 
<span id="IEq79"><mml:math id="IEq79_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq79_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq79.gif"/></span>
 increases. On the other hand, when 
<span id="IEq80"><mml:math id="IEq80_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq80_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq80.gif"/></span>
 increases Kurtosis increases.</p></section>
<section><h2>Order statistics</h2>
<p>Let 
<span id="IEq81"><mml:math id="IEq81_Math"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><tex-math id="IEq81_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Theta _{1},\Theta _{2},\ldots ,\Theta _{n}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq81.gif"/></span>
 be a random sample from 
<span id="IEq82"><mml:math id="IEq82_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq82_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq82.gif"/></span>
 distribution and let 
<span id="IEq83"><mml:math id="IEq83_Math"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>…</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math id="IEq83_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Theta _{(1)} \ldots \Theta _{(n)}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq83.gif"/></span>
, 
<span id="IEq84"><mml:math id="IEq84_Math"><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mo>⋯</mml:mo><mml:mo>&lt;</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mfenced></mml:math><tex-math id="IEq84_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\left( \Theta _{(1)}< \cdots <\Theta _{(n)}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq84.gif"/></span>
, denote the order statistic for this sample. Then, the pdf of the random variable 
<span id="IEq85"><mml:math id="IEq85_Math"><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:math><tex-math id="IEq85_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Theta _{(i)}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq85.gif"/></span>
, 
<span id="IEq86"><mml:math id="IEq86_Math"><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math><tex-math id="IEq86_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$i=1,2,\ldots ,n$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq86.gif"/></span>
 is obtained as
<section id="Equ18"><mml:math display="block" id="Equ18_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>!</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:mi>F</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mi>n</mml:mi><mml:mo>!</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>κ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>κ</mml:mi></mml:mfenced><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>!</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>κ</mml:mi><mml:mo>-</mml:mo><mml:mi>κ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup></mml:mfenced><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ18_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} f_{\Theta _{(i)}}\left( \theta \right)&=\frac{n!}{(i-1)!(n-i)!} F(\theta )^{i-1}f(\theta )(1-F(\theta ))^{n-i}\nonumber \\&=\frac{\lambda n!{\mathrm{e}}^{-\lambda \theta }\kappa ^{i-1}\left( \Lambda c-2\kappa \right) \left( {\mathrm{e}}^{-\lambda \theta }-1\right) ^{i-1}}{c^{2n} (i-1)!(n-i)!}\left( c^{2}+\kappa -\kappa {\mathrm{e}}^{-\lambda \theta }\right) ^{n-i}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ18.gif"/></section>
where, 
<span id="IEq87"><mml:math id="IEq87_Math"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:math><tex-math id="IEq87_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\kappa =\Lambda +c-\Lambda {\mathrm{e}}^{-\lambda \theta }+\Lambda c$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq87.gif"/></span>
. The first order and 
<em>n</em>
th order statistics can be immediately calculated from (
<a href="#Equ18"><sup>18</sup></a>
) as
<section id="Equ28"><mml:math display="block" id="Equ28_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>κ</mml:mi></mml:mfenced></mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mfrac><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>κ</mml:mi><mml:mo>-</mml:mo><mml:mi>κ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup></mml:mfenced><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ28_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} f_{\Theta _{(1)}}\left( \theta \right) =\frac{\lambda n{\mathrm{e}}^{-\lambda \theta }\left( \Lambda c-2\kappa \right) }{c^{2n}}\left( c^{2}+\kappa -\kappa {\mathrm{e}}^{-\lambda \theta }\right) ^{n-1}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ28.gif"/></section>
and
<section id="Equ29"><mml:math display="block" id="Equ29_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mi>n</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>κ</mml:mi></mml:mfenced></mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mfrac><mml:msup><mml:mi>κ</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ29_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} f_{\Theta _{(n)}}\left( \theta \right) =\frac{\lambda n{\mathrm{e}}^{-\lambda \theta }\left( \Lambda c-2\kappa \right) }{c^{2n}}\kappa ^{n-1}\left( {\mathrm{e}}^{-\lambda \theta }-1\right) ^{n-1}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ29.gif"/></section>
respectively.</p></section>
<section><h2>Rényi and Shannon entropy</h2>
<p>The entropy is a measure of variation or uncertainty of a random variable. In this section, we investigate the Shannon and Rényi entropy, which are two most popular entropies, for TWE distribution. The Rényi entropy of a circular random variable with pdf 
<span id="IEq88"><mml:math id="IEq88_Math"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><tex-math id="IEq88_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$f(\theta )$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq88.gif"/></span>
 is defined as
<section id="Equ30"><mml:math display="block" id="Equ30_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi mathvariant="normal">RE</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>ξ</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi></mml:mrow></mml:mfrac><mml:mo>ln</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mi>f</mml:mi><mml:mi>ξ</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>θ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ30_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} {\mathrm{RE}}_{\theta }\left( \xi \right) =\frac{1}{1-\xi }\ln \int _{0}^{2\pi }f^{\xi }(\theta ){\mathrm{d}}\theta , \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ30.gif"/></section>
for 
<span id="IEq89"><mml:math id="IEq89_Math"><mml:mrow><mml:mi>ξ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq89_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\xi >0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq89.gif"/></span>
 and 
<span id="IEq90"><mml:math id="IEq90_Math"><mml:mrow><mml:mi>ξ</mml:mi><mml:mo>≠</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq90_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\xi \ne 1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq90.gif"/></span>
. Thus, Rényi entropy of 
<span id="IEq91"><mml:math id="IEq91_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq91_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq91.gif"/></span>
 distribution is obtained as
<section id="Equ31"><mml:math display="block" id="Equ31_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>R</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>ξ</mml:mi></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>ξ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>ξ</mml:mi></mml:mrow></mml:mfrac><mml:mspace width="0.166667em"/><mml:mfenced close="" open="[" separators=""><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi><mml:mo>;</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mo>;</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mi>λ</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mrow><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi></mml:mrow></mml:msup></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mfenced close="]" open="" separators=""><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mrow><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mo>-</mml:mo><mml:mi>ξ</mml:mi><mml:mo>;</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>ξ</mml:mi><mml:mo>;</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac></mml:mfenced></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ31_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} RE_{\theta }\left( \xi \right)&=\frac{2^{\xi -1}}{\lambda (1-\xi )\xi }\,\left[ _{2}F_{1}\left( -2\xi ,-\xi ;1-2\xi ;\frac{\Lambda c+c+2\Lambda }{2\Lambda }\right) \left( \frac{c^{2}}{\lambda \Lambda }\right) ^{-\xi }\right. \\&\quad \left. -\,\left( \frac{c^{2}{\mathrm{e}}^{4\pi \lambda }}{\lambda \Lambda }\right) ^{-\xi }\,_{2}F_{1}\left( -2\xi ,-\xi ;1-2\xi ;\frac{{\mathrm{e}}^{2\pi \lambda }(\Lambda c+c+2\Lambda )}{2\Lambda }\right) \right] , \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ31.gif"/></section>
where 
<span id="IEq92"><mml:math id="IEq92_Math"><mml:mrow><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><tex-math id="IEq92_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$_{2}F_{1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq92.gif"/></span>
 denotes the hypergeometric function, see [
<a href="#CR1"><sup>1</sup></a>
]. The Shannon entropy is the special case of the Rényi entropy when 
<span id="IEq93"><mml:math id="IEq93_Math"><mml:mrow><mml:mi>ξ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq93_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\xi \rightarrow 1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq93.gif"/></span>
 and it is defined as 
<span id="IEq94"><mml:math id="IEq94_Math"><mml:mrow><mml:mi>S</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mfenced close="]" open="[" separators=""><mml:mo>-</mml:mo><mml:mo>ln</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq94_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$SE_{\theta }=E\left[ -\ln f(\theta )\right] $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq94.gif"/></span>
, see [
<a href="#CR11"><sup>11</sup></a>
] for definition of Shannon entropy. Immediately, Shannon entropy of 
<span id="IEq95"><mml:math id="IEq95_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq95_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq95.gif"/></span>
 distribution is obtained as
<section id="Equ32"><mml:math display="block" id="Equ32_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>S</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>θ</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac><mml:mfenced close="" open="[" separators=""><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>+</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mi>λ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mfenced close="]" open="" separators=""><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mfenced></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ32_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} SE_{\theta }&=\frac{1}{4\left( {\mathrm{e}}^{2\pi \lambda }-1\right) ^{2}\Lambda }\left[ \left( {\mathrm{e}}^{2\pi \lambda }(\Lambda -1)+\Lambda +1\right) ^{2}\ln \left( \frac{1-\Lambda )}{\Lambda +1}\right) \right. \\&\quad +\,2{\mathrm{e}}^{2\pi \lambda }\Lambda \left( -{\mathrm{e}}^{2\pi \lambda }(\Lambda -3)-2\left( {\mathrm{e}}^{2\pi \lambda }-\Lambda -1\right) \ln \left( \frac{{\mathrm{e}}^{2\pi \lambda } \lambda (\Lambda +1)}{{\mathrm{e}}^{2\pi \lambda }-1}\right) -3(\Lambda +1)\right) \\&\quad \left. -\,2\Lambda \left( -3{\mathrm{e}}^{2\pi \lambda }(\Lambda -1)+2\left( {\mathrm{e}}^{2\pi \lambda }(\Lambda -1)+1\right) \ln \left( -\frac{\lambda (\Lambda -1)}{{\mathrm{e}}^{2\pi \lambda }-1}\right) -\Lambda -3\right) \right] . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ32.gif"/></section></p></section>
<section><h2>Stress-strength reliability</h2>
<p>Suppose 
<em>Y</em>
 represents the ‘stress’ and 
<em>X</em>
 represents the ‘strength’ to sustain the stress, then the stress-strength reliability is denoted by 
<span id="IEq96"><mml:math id="IEq96_Math"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>Y</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>X</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq96_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$R=P\left( Y<X\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq96.gif"/></span>
. Let 
<span id="IEq97"><mml:math id="IEq97_Math"><mml:mrow><mml:mi>X</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq97_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$X\sim {\mathrm{TWE}}\left( \lambda _{x},\Lambda _{x}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq97.gif"/></span>
 and 
<span id="IEq98"><mml:math id="IEq98_Math"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq98_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$Y\sim {\mathrm{TWE}}\left( \lambda _{y},\Lambda _{y}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq98.gif"/></span>
. Stress-strength reliability 
<span id="IEq99"><mml:math id="IEq99_Math"><mml:mrow><mml:mi>P</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>Y</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>X</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq99_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$P\left( Y<X\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq99.gif"/></span>
 is
<section id="Equ33"><mml:math display="block" id="Equ33_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mi>R</mml:mi></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>Y</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>X</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mo>∫</mml:mo><mml:mi>P</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>Y</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>X</mml:mi><mml:mspace width="3.33333pt"/><mml:mo stretchy="false">|</mml:mo><mml:mspace width="3.33333pt"/><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi></mml:mfenced><mml:msub><mml:mi>f</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mo>∫</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:msub><mml:mi>F</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>x</mml:mi></mml:mfenced><mml:mi mathvariant="normal">d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mfrac><mml:mfenced close="" open="[" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mfenced></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfenced></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>+</mml:mo><mml:mspace width="0.166667em"/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>+</mml:mo><mml:mspace width="0.166667em"/><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mfenced close="]" open="" separators=""><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mfrac><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ33_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} R&=P\left( Y<X\right) =\int P\left( Y<X~|~X=x\right) f_{X}\left( x\right) {\mathrm{d}}x=\int f_{X}\left( x\right) F_{Y}\left( x\right) {\mathrm{d}}x\\&=\frac{1}{c_{x}^{2}c_{y}^{2}}\left[ \Lambda _{y}{\mathrm{e}}^{-2\pi \left( \lambda _{x}+2\lambda _{y}\right) }\left( \frac{{\mathrm{e}}^{2\pi \lambda _{x}}\Lambda _{x} }{\lambda _{y}}-\frac{\left( c_{x}+2\right) \Lambda _{x}+c_{x}}{\lambda _{x}+2\lambda _{y}}\right) \right. \\&\quad -\,4\pi \Lambda _{x}\left( c_{y}\left( \Lambda _{y}+1\right) +\Lambda _{y}\right) +\frac{\Lambda _{y}\left( c_{x}\left( \Lambda _{x}+1\right) \lambda _{y}-\lambda _{x}\Lambda _{x}\right) }{\lambda _{y}\left( \lambda _{x}+2\lambda _{y}\right) }\\&\quad +\,\frac{2\Lambda _{x}{\mathrm{e}}^{-2\pi \lambda _{y}}\left( {\mathrm{e}}^{2\pi \lambda _{y} }-1\right) \left( \left( c_{y}+2\right) \Lambda _{y}+c_{y}\right) }{\lambda _{x}}\\&\quad +\,\frac{{\mathrm{e}}^{-2\pi \lambda _{x}}\left( {\mathrm{e}}^{2\pi \lambda _{x}}-1\right) \left( \left( c_{x}+2\right) \Lambda _{x}+c_{x}\right) \left( c_{y}\left( \Lambda _{y}+1\right) +\Lambda _{y}\right) }{\lambda _{x}}\\&\quad \left. -\,\frac{\left( \left( c_{x}+2\right) \Lambda _{x}+c_{x}\right) \left( \left( c_{y}+2\right) \Lambda _{y}+c_{y}\right) {\mathrm{e}}^{-2\pi \left( \lambda _{x}+\lambda _{y}\right) }\left( {\mathrm{e}}^{2\pi \left( \lambda _{x} +\lambda _{y}\right) }+1\right) }{\lambda _{x}+\lambda _{y}}\right] , \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ33.gif"/></section>
where 
<span id="IEq100"><mml:math id="IEq100_Math"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq100_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$c_{x}={\mathrm{e}}^{-2\pi \lambda _{x}}-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq100.gif"/></span>
 and 
<span id="IEq101"><mml:math id="IEq101_Math"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq101_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$c_{y}={\mathrm{e}}^{-2\pi \lambda _{y}}-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq101.gif"/></span>
. If 
<span id="IEq102"><mml:math id="IEq102_Math"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:math><tex-math id="IEq102_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda _{x}=\lambda _{y}=\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq102.gif"/></span>
<section id="Equ34"><mml:math display="block" id="Equ34_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mfenced close="]" open="[" separators=""><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>6</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mo>+</mml:mo><mml:mspace width="0.166667em"/><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>24</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mn>15</mml:mn></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>3</mml:mn><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>12</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mfrac><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mfenced><mml:mn>6</mml:mn></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ34_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} R=\frac{\Lambda _{x}}{6\left( {\mathrm{e}}^{2\pi \lambda }-1\right) ^{4}}\left[ \begin{array}{c} -\,6{\mathrm{e}}^{4\pi \lambda }\left( \Lambda _{y}+2\right) -\,2{\mathrm{e}}^{2\pi \lambda }\left( \Lambda _{y}+3\right) \\ +\,{\mathrm{e}}^{8\pi \lambda }\left( 24\pi \lambda +5\Lambda _{y}-15\right) +\Lambda _{y}+3\\ -\,2{\mathrm{e}}^{6\pi \lambda }\left( 3(4\pi \lambda -5)+(12\pi \lambda -1)\Lambda _{y}\right) \end{array} \right] +\frac{\left( \Lambda _{y}+3\right) }{6}. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ34.gif"/></section></p></section>
<section><h2>Hazard rate function</h2>
<p>The hazard rate function 
<span id="IEq103"><mml:math id="IEq103_Math"><mml:msub><mml:mi>h</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math><tex-math id="IEq103_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$h_{r}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq103.gif"/></span>
 of 
<span id="IEq104"><mml:math id="IEq104_Math"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq104_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Theta \sim {\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq104.gif"/></span>
 random variable is
<section id="Equ35"><mml:math display="block" id="Equ35_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>F</mml:mi><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>c</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ35_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} h_{r}\left( \theta \right)&=\frac{f\left( \theta \right) }{1-F\left( \theta \right) }\\&=\frac{\lambda {\mathrm{e}}^{-\theta \lambda }\left( 2\Lambda +c-2\Lambda {\mathrm{e}}^{-\theta \lambda }+\Lambda c\right) }{\left( \Lambda +c-\Lambda {\mathrm{e}}^{-\theta \lambda }\right) \left( {\mathrm{e}}^{-\theta \lambda }-c-1\right) }, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ35.gif"/></section>
where 
<span id="IEq105"><mml:math id="IEq105_Math"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mfenced close=")" open="[" separators=""><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width="3.33333pt"/><mml:mi>λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq105_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\theta \in \left[ 0,2\pi \right) ,~\lambda >0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq105.gif"/></span>
, 
<span id="IEq106"><mml:math id="IEq106_Math"><mml:mrow><mml:mfenced close="|" open="|"><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq106_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\left| \Lambda \right| \le 1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq106.gif"/></span>
 and 
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				\begin{document}$$c={\mathrm{e}}^{-2\pi \lambda }-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq107.gif"/></span>
. Critical point of the 
<span id="IEq108"><mml:math id="IEq108_Math"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq108_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$h_{r}\left( \theta \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq108.gif"/></span>
 is
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				\begin{document}$$\begin{aligned} \theta _{h}=\frac{1}{\lambda }\ln \left[ \frac{\sqrt{-c^{2}(c+1)(\Lambda -1)^{2}\Lambda (c+\Lambda )}+2(c+1)\Lambda (c+\Lambda )}{(c+1)(c+\Lambda )((c+2)\Lambda +c)}\right] . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ36.gif"/></section>
The hazard rate function has bathtub shape when 
<span id="IEq109"><mml:math id="IEq109_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq109_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq109.gif"/></span>
 is in the interval
<section id="Equ37"><mml:math display="block" id="Equ37_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfenced close="]" open="[" separators=""><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:msqrt></mml:mfenced><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfenced close="]" open="[" separators=""><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:msqrt></mml:mfenced></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ37_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} \left( \frac{1}{2}\left[ -c-\sqrt{c^{2}-4\left( c+1\right) }\right] ,\frac{1}{2}\left[ -c+\sqrt{c^{2}-4\left( c+1\right) }\right] \right) . \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ37.gif"/></section>
Here, considering that the smallest value of 
<em>c</em>
 is 
<span id="IEq110"><mml:math id="IEq110_Math"><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq110_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq110.gif"/></span>
, 
<span id="IEq111"><mml:math id="IEq111_Math"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi>θ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq111_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$h_{r}\left( \theta \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq111.gif"/></span>
 appears to be a bathtub in the positive 
<span id="IEq112"><mml:math id="IEq112_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq112_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq112.gif"/></span>
 values providing the above condition. We present Fig. 
<a href="#Fig3"><sup>3</sup></a>
 which plots the hazard rate functions of the 
<span id="IEq113"><mml:math id="IEq113_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>5.48</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.277778em"/><mml:mn>0.25</mml:mn></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq113_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( 5.48,\; 0.25\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq113.gif"/></span>
 and WE
<span id="IEq114"><mml:math id="IEq114_Math"><mml:mfenced close=")" open="(" separators=""><mml:mn>5.48</mml:mn></mml:mfenced></mml:math><tex-math id="IEq114_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\left( 5.48\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq114.gif"/></span>
 distributions for illustrative purposes.
<figure id="Fig3"><h3>Fig. 3</h3>
<figcaption><p>The plots of hazard rate functions of TWE and WE distributions</p></figcaption>
<img src="40096_2018_268_Fig3_HTML.png" /></figure></p></section></section>
<section><h2>Inference</h2>
<p>In this section, we consider the statistical inference problem for 
<span id="IEq115"><mml:math id="IEq115_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq115_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq115.gif"/></span>
. To estimate the unknown parameters of 
<span id="IEq116"><mml:math id="IEq116_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq116_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq116.gif"/></span>
, we employ the ML, LS and WLS estimation methods commonly used in the literature.</p>
<section><h2>Maximum likelihood estimation</h2>
<p>Let 
<span id="IEq117"><mml:math id="IEq117_Math"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><tex-math id="IEq117_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\Theta _{1},\Theta _{2},\ldots ,\Theta _{n}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq117.gif"/></span>
 be a random sample from 
<span id="IEq118"><mml:math id="IEq118_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq118_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq118.gif"/></span>
 distribution. From (
<a href="#Equ6"><sup>6</sup></a>
), the logarithmic likelihood function for the random variables 
<span id="IEq119"><mml:math id="IEq119_Math"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math><tex-math id="IEq119_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Theta _{i},i=1,2,\ldots ,n$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq119.gif"/></span>
 can be immediately written as
<section id="Equ19"><mml:math display="block" id="Equ19_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>L</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>;</mml:mo><mml:mspace width="0.277778em"/><mml:msub><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mo>ln</mml:mo><mml:mi>λ</mml:mi><mml:mo>-</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mspace width="1em"/><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mo>ln</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>-</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ19_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} L\left( \lambda ,\Lambda ;\; \theta _{1},\theta _{2},\ldots ,\theta _{n}\right)&=\sum _{i=1}^{n}\ln \lambda -\sum _{i=1}^{n}\ln \left( 1-{\mathrm{e}}^{-2\pi \lambda }\right) \nonumber \\&\quad +\sum _{i=1}^{n}\ln \left( \Lambda -\frac{2\Lambda \left( {\mathrm{e}}^{-\lambda \theta _{i}}-1\right) }{{\mathrm{e}}^{-2\pi \lambda }-1}+1\right) -\sum _{i=1}^{n} \lambda \theta _{i} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ19.gif"/></section>
If the first derivatives of (
<a href="#Equ19"><sup>19</sup></a>
) with respect to parameters 
<span id="IEq120"><mml:math id="IEq120_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq120_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq120.gif"/></span>
 and 
<span id="IEq121"><mml:math id="IEq121_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq121_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq121.gif"/></span>
 are taken and equalized them to zero, then we have the following normal equations
<section id="Equ20"><mml:math display="block" id="Equ20_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mi>n</mml:mi><mml:mi>λ</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mi>c</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mi>c</mml:mi></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>π</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>-</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ20_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \frac{\partial L}{\partial \lambda }=\frac{n}{\lambda }+\frac{2n\pi {\mathrm{e}}^{-2\pi \lambda }}{c}+\sum _{i=1}^{n}\frac{\frac{2\Lambda \theta _{i} {\mathrm{e}}^{-\lambda \theta _{i}}}{c}-\frac{4\Lambda \pi \left( C+1\right) {\mathrm{e}}^{-\lambda \theta _{i}}}{c^{2}}}{c\left( \Lambda +1\right) -\left( 2\Lambda \left( {\mathrm{e}}^{-\lambda \theta _{i}}-1\right) \right) }-\sum _{i=1}^{n}\theta _{i}=0 \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ20.gif"/></section>
and
<section id="Equ21"><mml:math display="block" id="Equ21_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mo>-</mml:mo><mml:mfenced close=")" open="(" separators=""><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ21_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} \frac{\partial L}{\partial \Lambda }=\sum _{i=1}^{n}\frac{c-\left( 2{\mathrm{e}}^{-\lambda \theta _{i}}-2\right) }{c\left( \Lambda +1\right) -\left( 2\Lambda \left( {\mathrm{e}}^{-\lambda \theta _{i}}-1\right) \right) }=0 \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ21.gif"/></section>
where 
<span id="IEq122"><mml:math id="IEq122_Math"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq122_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$c={\mathrm{e}}^{-2\pi \lambda }-1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq122.gif"/></span>
. Hence, the ML estimates of the parameters 
<span id="IEq123"><mml:math id="IEq123_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq123_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq123.gif"/></span>
 and 
<span id="IEq124"><mml:math id="IEq124_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq124_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq124.gif"/></span>
, say 
<span id="IEq125"><mml:math id="IEq125_Math"><mml:msub><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover><mml:mi mathvariant="normal">ML</mml:mi></mml:msub></mml:math><tex-math id="IEq125_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\lambda }}_{{\mathrm{ML}}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq125.gif"/></span>
 and 
<span id="IEq126"><mml:math id="IEq126_Math"><mml:msub><mml:mover accent="true"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover><mml:mi mathvariant="normal">ML</mml:mi></mml:msub></mml:math><tex-math id="IEq126_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\Lambda }}_{\mathrm{ML}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq126.gif"/></span>
, respectively, can numerically be obtained from the collective solution of (
<a href="#Equ20"><sup>20</sup></a>
) and (
<a href="#Equ21"><sup>21</sup></a>
).</p></section>
<section><h2>Least squares estimation</h2>
<p>To obtain the least squares estimates of the 
<span id="IEq127"><mml:math id="IEq127_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq127_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq127.gif"/></span>
 distribution, let us consider the ordered random sample 
<span id="IEq128"><mml:math id="IEq128_Math"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mo>⋯</mml:mo><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math id="IEq128_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\theta _{(1)}< \cdots <\theta _{(n)}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq128.gif"/></span>
 from this distribution. Then, the LS estimates of the unknown parameters of the 
<span id="IEq129"><mml:math id="IEq129_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq129_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq129.gif"/></span>
 distribution, say 
<span id="IEq130"><mml:math id="IEq130_Math"><mml:msub><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover><mml:mrow><mml:mi mathvariant="italic">LS</mml:mi></mml:mrow></mml:msub></mml:math><tex-math id="IEq130_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\lambda }}_{LS}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq130.gif"/></span>
, and 
<span id="IEq131"><mml:math id="IEq131_Math"><mml:msub><mml:mover accent="true"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover><mml:mrow><mml:mi mathvariant="italic">LS</mml:mi></mml:mrow></mml:msub></mml:math><tex-math id="IEq131_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\Lambda }}_{LS}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq131.gif"/></span>
, are obtained by minimizing
<section id="Equ38"><mml:math display="block" id="Equ38_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mfrac><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mfenced close=")" open="("><mml:mi>j</mml:mi></mml:mfenced></mml:msub></mml:mrow></mml:msup></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mfenced close=")" open="("><mml:mi>j</mml:mi></mml:mfenced></mml:msub><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mfenced></mml:mfenced></mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ38_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} \sum _{j=1}^{n}\left( \frac{\left( 1-{\mathrm{e}}^{-\lambda \theta _{\left( j\right) } }\right) \left( c+\Lambda \left( 1+c-{\mathrm{e}}^{-\theta _{\left( j\right) }\lambda }\right) \right) }{c^{2}}-\frac{j}{n+1}\right) ^{2}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ38.gif"/></section>
with respect to 
<span id="IEq132"><mml:math id="IEq132_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq132_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq132.gif"/></span>
 and 
<span id="IEq133"><mml:math id="IEq133_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq133_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq133.gif"/></span>
, respectively. Where 
<span id="IEq134"><mml:math id="IEq134_Math"><mml:mfrac><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:math><tex-math id="IEq134_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\frac{j}{n+1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq134.gif"/></span>
 is the expectation of the empirical distribution function of the ordered data, see Swain et al. [
<a href="#CR19"><sup>19</sup></a>
]. It is known that the LS estimates are biassed. A well-known modification of LS method is the WLS, which has a lower bias than the ordinary LS. The WLS estimates of the parameters of the 
<span id="IEq135"><mml:math id="IEq135_Math"><mml:mrow><mml:mi mathvariant="normal">TWE</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:mfenced></mml:mrow></mml:math><tex-math id="IEq135_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\mathrm{TWE}}\left( \lambda ,\Lambda \right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq135.gif"/></span>
 distribution are obtained by minimizing
<section id="Equ39"><mml:math display="block" id="Equ39_Math"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=")" open="(" separators=""><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:msup><mml:mfenced close="]" open="[" separators=""><mml:mfrac><mml:mrow><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mfenced close=")" open="("><mml:mi>j</mml:mi></mml:mfenced></mml:msub></mml:mrow></mml:msup></mml:mfenced><mml:mfenced close=")" open="(" separators=""><mml:mi>c</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mfenced close=")" open="("><mml:mi>j</mml:mi></mml:mfenced></mml:msub><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:mfenced></mml:mfenced></mml:mrow><mml:msup><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ39_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \sum _{j=1}^{n}\frac{\left( n+1\right) ^{2}\left( n+2\right) }{j\left( n-j+1\right) }\left[ \frac{\left( 1-{\mathrm{e}}^{-\lambda \theta _{\left( j\right) } }\right) \left( c+\Lambda \left( 1+c-{\mathrm{e}}^{-\theta _{\left( j\right) }\lambda }\right) \right) }{c^{2}}-\frac{j}{n+1}\right] ^{2}, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_Equ39.gif"/></section>
with respect to 
<span id="IEq136"><mml:math id="IEq136_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq136_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq136.gif"/></span>
 and 
<span id="IEq137"><mml:math id="IEq137_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq137_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq137.gif"/></span>
.</p></section></section>
<section><h2>Monte Carlo simulation study</h2>
<p>In this section, we perform some Monte Carlo simulation studies for illustrating and comparing estimation performances of the ML, LS and the WLS estimators obtained in the previous section. In Monte Carlo simulations, we use the values of the parameters 
<span id="IEq138"><mml:math id="IEq138_Math"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mo>,</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq138_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda =0.5,1.5$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq138.gif"/></span>
 and 
<span id="IEq139"><mml:math id="IEq139_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>0.75</mml:mn><mml:mo>,</mml:mo><mml:mn>0.75</mml:mn></mml:mrow></mml:math><tex-math id="IEq139_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda =-\,0.75,0.75$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq139.gif"/></span>
. For the different sample of sizes 
<span id="IEq140"><mml:math id="IEq140_Math"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>30</mml:mn><mml:mo>,</mml:mo><mml:mn>50</mml:mn><mml:mo>,</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math><tex-math id="IEq140_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$n = 30, 50,100$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq140.gif"/></span>
 and 1000, the obtained Bias and mean squared error (MSE) values based on the 1000 times replicated simulations are displayed in Table 
<a href="#Tab1"><sup>1</sup></a>
.
<figure id="Tab1"><h3>Table 1</h3>
<figcaption><p>Bias and MSE of parameter estimations for different values of sample of sizes 
<em>n</em>
 and parameter 
<span id="IEq141"><mml:math id="IEq141_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq141_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq141.gif"/></span>
, when 
<span id="IEq142"><mml:math id="IEq142_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>0.75</mml:mn></mml:mrow></mml:math><tex-math id="IEq142_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\Lambda =-\,0.75$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq142.gif"/></span>
.and 0.75</p></figcaption>
<table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="3"/><th align="left" rowspan="3"><p>Method</p></th><th char="." align="left" rowspan="3"><p><italic>n</italic></p></th><th align="left" colspan="4"><p><inline-formula id="IEq143"><alternatives><mml:math id="IEq143_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mn>0.75</mml:mn></mml:mrow></mml:math><tex-math id="IEq143_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda =-\,0.75$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq143.gif"/></alternatives></inline-formula></p></th><th align="left" colspan="4"><p><inline-formula id="IEq144"><alternatives><mml:math id="IEq144_Math"><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.75</mml:mn></mml:mrow></mml:math><tex-math id="IEq144_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\usepackage{mathrsfs}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\Lambda =0.75$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq144.gif"/></alternatives></inline-formula></p></th></tr><tr><th char="." align="left" colspan="2"><p><inline-formula id="IEq145"><alternatives><mml:math id="IEq145_Math"><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math><tex-math id="IEq145_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
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				\begin{document}$${\hat{\lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq145.gif"/></alternatives></inline-formula></p></th><th align="left" colspan="2"><p><inline-formula id="IEq146"><alternatives><mml:math id="IEq146_Math"><mml:mover accent="true"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math><tex-math id="IEq146_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\Lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq146.gif"/></alternatives></inline-formula></p></th><th align="left" colspan="2"><p><inline-formula id="IEq147"><alternatives><mml:math id="IEq147_Math"><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math><tex-math id="IEq147_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\hat{\lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq147.gif"/></alternatives></inline-formula></p></th><th align="left" colspan="2"><p><inline-formula id="IEq148"><alternatives><mml:math id="IEq148_Math"><mml:mover accent="true"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math><tex-math id="IEq148_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\begin{document}$${\hat{\Lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq148.gif"/></alternatives></inline-formula></p></th></tr><tr><th align="left"><p>Bias</p></th><th align="left"><p>MSE</p></th><th align="left"><p>Bias</p></th><th align="left"><p>MSE</p></th><th align="left"><p>Bias</p></th><th align="left"><p>MSE</p></th><th align="left"><p>Bias</p></th><th align="left"><p>MSE</p></th></tr></thead><tbody><tr><td align="left" rowspan="12"><p><inline-formula id="IEq149"><alternatives><mml:math id="IEq149_Math"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq149_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda =0.5$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq149.gif"/></alternatives></inline-formula></p></td><td align="left" rowspan="4"><p>ML</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.1204</p></td><td char="." align="char"><p>0.0235</p></td><td char="." align="char"><p>0.2770</p></td><td char="." align="char"><p>0.1357</p></td><td char="." align="char"><p>0.1385</p></td><td char="." align="char"><p>0.0303</p></td><td char="." align="char"><p>0.2390</p></td><td char="." align="char"><p>0.0664</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.0910</p></td><td char="." align="char"><p>0.0145</p></td><td char="." align="char"><p>0.2150</p></td><td char="." align="char"><p>0.0834</p></td><td char="." align="char"><p>0.1208</p></td><td char="." align="char"><p>0.0222</p></td><td char="." align="char"><p>0.2259</p></td><td char="." align="char"><p>0.0611</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.0621</p></td><td char="." align="char"><p>0.0064</p></td><td char="." align="char"><p>0.1511</p></td><td char="." align="char"><p>0.0401</p></td><td char="." align="char"><p>0.1073</p></td><td char="." align="char"><p>0.0169</p></td><td char="." align="char"><p>0.2120</p></td><td char="." align="char"><p>0.0577</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0193</p></td><td char="." align="char"><p>0.0006</p></td><td char="." align="char"><p>0.0475</p></td><td char="." align="char"><p>0.0035</p></td><td char="." align="char"><p>0.0646</p></td><td char="." align="char"><p>0.0079</p></td><td char="." align="char"><p>0.1294</p></td><td char="." align="char"><p>0.0311</p></td></tr><tr><td align="left" rowspan="4"><p>WLS</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.1436</p></td><td char="." align="char"><p>0.0362</p></td><td char="." align="char"><p>0.3621</p></td><td char="." align="char"><p>0.2653</p></td><td char="." align="char"><p>0.1953</p></td><td char="." align="char"><p>0.0761</p></td><td char="." align="char"><p>0.3504</p></td><td char="." align="char"><p>0.2056</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.1157</p></td><td char="." align="char"><p>0.0283</p></td><td char="." align="char"><p>0.2991</p></td><td char="." align="char"><p>0.2188</p></td><td char="." align="char"><p>0.1602</p></td><td char="." align="char"><p>0.0472</p></td><td char="." align="char"><p>0.2950</p></td><td char="." align="char"><p>0.1405</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.0914</p></td><td char="." align="char"><p>0.0231</p></td><td char="." align="char"><p>0.2474</p></td><td char="." align="char"><p>0.1981</p></td><td char="." align="char"><p>0.1272</p></td><td char="." align="char"><p>0.0250</p></td><td char="." align="char"><p>0.2403</p></td><td char="." align="char"><p>0.0801</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0217</p></td><td char="." align="char"><p>0.0016</p></td><td char="." align="char"><p>0.0564</p></td><td char="." align="char"><p>0.0130</p></td><td char="." align="char"><p>0.0722</p></td><td char="." align="char"><p>0.0095</p></td><td char="." align="char"><p>0.1426</p></td><td char="." align="char"><p>0.0364</p></td></tr><tr><td align="left" rowspan="4"><p>LS</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.1329</p></td><td char="." align="char"><p>0.0284</p></td><td char="." align="char"><p>0.3291</p></td><td char="." align="char"><p>0.1906</p></td><td char="." align="char"><p>0.3201</p></td><td char="." align="char"><p>0.1718</p></td><td char="." align="char"><p>0.5798</p></td><td char="." align="char"><p>0.5220</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.0983</p></td><td char="." align="char"><p>0.0166</p></td><td char="." align="char"><p>0.2472</p></td><td char="." align="char"><p>0.1165</p></td><td char="." align="char"><p>0.3015</p></td><td char="." align="char"><p>0.1579</p></td><td char="." align="char"><p>0.5014</p></td><td char="." align="char"><p>0.3966</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.0662</p></td><td char="." align="char"><p>0.0076</p></td><td char="." align="char"><p>0.1729</p></td><td char="." align="char"><p>0.0549</p></td><td char="." align="char"><p>0.2502</p></td><td char="." align="char"><p>0.0977</p></td><td char="." align="char"><p>0.4650</p></td><td char="." align="char"><p>0.3229</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0165</p></td><td char="." align="char"><p>0.0005</p></td><td char="." align="char"><p>0.0343</p></td><td char="." align="char"><p>0.0032</p></td><td char="." align="char"><p>0.1703</p></td><td char="." align="char"><p>0.0538</p></td><td char="." align="char"><p>0.3290</p></td><td char="." align="char"><p>0.2001</p></td></tr><tr><td align="left" rowspan="12"><p><inline-formula id="IEq150"><alternatives><mml:math id="IEq150_Math"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq150_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda =1.5$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq150.gif"/></alternatives></inline-formula></p></td><td align="left" rowspan="4"><p>ML</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.2251</p></td><td char="." align="char"><p>0.0862</p></td><td char="." align="char"><p>0.2433</p></td><td char="." align="char"><p>0.1127</p></td><td char="." align="char"><p>0.3415</p></td><td char="." align="char"><p>0.1929</p></td><td char="." align="char"><p>0.2337</p></td><td char="." align="char"><p>0.0643</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.1733</p></td><td char="." align="char"><p>0.0493</p></td><td char="." align="char"><p>0.1903</p></td><td char="." align="char"><p>0.0682</p></td><td char="." align="char"><p>0.3088</p></td><td char="." align="char"><p>0.1474</p></td><td char="." align="char"><p>0.2268</p></td><td char="." align="char"><p>0.0639</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.1178</p></td><td char="." align="char"><p>0.0223</p></td><td char="." align="char"><p>0.1304</p></td><td char="." align="char"><p>0.0266</p></td><td char="." align="char"><p>0.2654</p></td><td char="." align="char"><p>0.1097</p></td><td char="." align="char"><p>0.2089</p></td><td char="." align="char"><p>0.0585</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0385</p></td><td char="." align="char"><p>0.0023</p></td><td char="." align="char"><p>0.0431</p></td><td char="." align="char"><p>0.0029</p></td><td char="." align="char"><p>0.1632</p></td><td char="." align="char"><p>0.0440</p></td><td char="." align="char"><p>0.1357</p></td><td char="." align="char"><p>0.0293</p></td></tr><tr><td align="left" rowspan="4"><p>WLS</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.2586</p></td><td char="." align="char"><p>0.1177</p></td><td char="." align="char"><p>0.2996</p></td><td char="." align="char"><p>0.2118</p></td><td char="." align="char"><p>0.4908</p></td><td char="." align="char"><p>0.5281</p></td><td char="." align="char"><p>0.3477</p></td><td char="." align="char"><p>0.2213</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.1948</p></td><td char="." align="char"><p>0.0689</p></td><td char="." align="char"><p>0.2265</p></td><td char="." align="char"><p>0.1238</p></td><td char="." align="char"><p>0.4366</p></td><td char="." align="char"><p>0.3930</p></td><td char="." align="char"><p>0.3202</p></td><td char="." align="char"><p>0.1739</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.1276</p></td><td char="." align="char"><p>0.0277</p></td><td char="." align="char"><p>0.1437</p></td><td char="." align="char"><p>0.0401</p></td><td char="." align="char"><p>0.3306</p></td><td char="." align="char"><p>0.1847</p></td><td char="." align="char"><p>0.2588</p></td><td char="." align="char"><p>0.0991</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0398</p></td><td char="." align="char"><p>0.0025</p></td><td char="." align="char"><p>0.0459</p></td><td char="." align="char"><p>0.0034</p></td><td char="." align="char"><p>0.1888</p></td><td char="." align="char"><p>0.0546</p></td><td char="." align="char"><p>0.1530</p></td><td char="." align="char"><p>0.0343</p></td></tr><tr><td align="left" rowspan="4"><p>LS</p></td><td char="." align="char"><p>30</p></td><td char="." align="char"><p>0.2547</p></td><td char="." align="char"><p>0.1047</p></td><td char="." align="char"><p>0.2859</p></td><td char="." align="char"><p>0.1587</p></td><td char="." align="char"><p>0.7485</p></td><td char="." align="char"><p>1.0383</p></td><td char="." align="char"><p>0.5199</p></td><td char="." align="char"><p>0.4485</p></td></tr><tr><td char="." align="char"><p>50</p></td><td char="." align="char"><p>0.1942</p></td><td char="." align="char"><p>0.0616</p></td><td char="." align="char"><p>0.2226</p></td><td char="." align="char"><p>0.0901</p></td><td char="." align="char"><p>0.6971</p></td><td char="." align="char"><p>0.8684</p></td><td char="." align="char"><p>0.5079</p></td><td char="." align="char"><p>0.4217</p></td></tr><tr><td char="." align="char"><p>100</p></td><td char="." align="char"><p>0.1378</p></td><td char="." align="char"><p>0.0305</p></td><td char="." align="char"><p>0.1566</p></td><td char="." align="char"><p>0.0409</p></td><td char="." align="char"><p>0.5153</p></td><td char="." align="char"><p>0.4557</p></td><td char="." align="char"><p>0.4128</p></td><td char="." align="char"><p>0.2737</p></td></tr><tr><td char="." align="char"><p>1000</p></td><td char="." align="char"><p>0.0370</p></td><td char="." align="char"><p>0.0025</p></td><td char="." align="char"><p>0.0408</p></td><td char="." align="char"><p>0.0034</p></td><td char="." align="char"><p>0.4361</p></td><td char="." align="char"><p>0.2897</p></td><td char="." align="char"><p>0.3325</p></td><td char="." align="char"><p>0.1615</p></td></tr></tbody></table></figure></p>
<p>As can be clearly seen from Table 
<a href="#Tab1"><sup>1</sup></a>
, when the sample size increases, for all values of the parameters, both Bias and MSE values decreases. This shows that the estimates are precise and accurate and hence consistent and unbiased. This is an expected result for the ML estimators, since ML estimators are asymptotically unbiased estimators. The simulation results also show that the other estimators have the same characteristics. Besides, by Table 
<a href="#Tab1"><sup>1</sup></a>
, we can say that the ML estimators outperform both the LS and the WLS estimators with smaller MSE values.</p></section>
<section><h2>Application to real data</h2>
<p>In this section, to illustrate the modeling behavior of the TWE distribution on a real-life dataset , we analyze the turtle dataset, which is a popular circular dataset. This dataset contains the orientations of 76 turtles laying their eggs [
<a href="#CR6"><sup>6</sup></a>
]. We obtain the maximum likelihood estimation of the parameters 
<span id="IEq151"><mml:math id="IEq151_Math"><mml:mi>λ</mml:mi></mml:math><tex-math id="IEq151_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq151.gif"/></span>
 and 
<span id="IEq152"><mml:math id="IEq152_Math"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math><tex-math id="IEq152_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 by using the “mle” subroutine in the package ‘stats4’ (version 3.4.3) of R. Note that when applying the mle subroutine, the parameter ranges should be selected as wide as possible to avoid local maxima. We also refer the advanced readers to an R package ‘wrapped’, introduced by Nadarajah and Zhang [
<a href="#CR15"><sup>15</sup></a>
], for further computation in wrapped distributions.</p>
<p>For the turtle dataset, the ML estimation of the parameters and the corresponding mean direction and the resultant length are obtained as given in Table 
<a href="#Tab2"><sup>2</sup></a>
, when the dataset is modeled by the TWE distribution.
<figure id="Tab2"><h3>Table 2</h3>
<figcaption><p>ML estimates for turtle data</p></figcaption>
<table frame="hsides" rules="groups"><thead><tr><th align="left"><p><inline-formula id="IEq153"><alternatives><mml:math id="IEq153_Math"><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:math><tex-math id="IEq153_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\widehat{\lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq153.gif"/></alternatives></inline-formula></p></th><th align="left"><p><inline-formula id="IEq154"><alternatives><mml:math id="IEq154_Math"><mml:mover accent="true"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:math><tex-math id="IEq154_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${\widehat{\Lambda }}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq154.gif"/></alternatives></inline-formula></p></th><th align="left"><p>Mean direction</p></th><th align="left"><p>Res. length</p></th></tr></thead><tbody><tr><td align="left"><p>0.7475</p></td><td align="left"><p><inline-formula id="IEq155"><alternatives><mml:math id="IEq155_Math"><mml:mo>-</mml:mo></mml:math><tex-math id="IEq155_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$-$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq155.gif"/></alternatives></inline-formula> 0.9513</p></td><td align="left"><p>1.49 <inline-formula id="IEq156"><alternatives><mml:math id="IEq156_Math"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∼</mml:mo><mml:mn>85.1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><tex-math id="IEq156_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
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				\begin{document}$$(\sim 85.1)$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq156.gif"/></alternatives></inline-formula>°</p></td><td align="left"><p>0.4978</p></td></tr></tbody></table></figure></p>
<p>This dataset was recently used by Joshi and Jose [
<a href="#CR7"><sup>7</sup></a>
] as an application of the wrapped Lindley 
<span id="IEq157"><mml:math id="IEq157_Math"><mml:mfenced close=")" open="("><mml:mi mathvariant="script">WL</mml:mi></mml:mfenced></mml:math><tex-math id="IEq157_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\left( {\mathcal {WL}}\right) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq157.gif"/></span>
 distribution. In order to make a comparison, maximized log likelihood values (L), Akaike information criterion (AIC), Kolmogorov–Smirnov with 
<em>p</em>
 values (KS) and Watson’s 
<span id="IEq158"><mml:math id="IEq158_Math"><mml:mrow><mml:msup><mml:mi>U</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mspace width="0.166667em"/><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mtext>W</mml:mtext><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq158_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$U^{2} \, (\hbox {W}^{2})$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq158.gif"/></span>
 statistics values for the TWE, WE and 
<span id="IEq159"><mml:math id="IEq159_Math"><mml:mi mathvariant="script">WL</mml:mi></mml:math><tex-math id="IEq159_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 distributions are given in Table 
<a href="#Tab3"><sup>3</sup></a>
.
<figure id="Tab3"><h3>Table 3</h3>
<figcaption><p>Summary of fits for turtle data</p></figcaption>
<table frame="hsides" rules="groups"><thead><tr><th align="left"><p>Model</p></th><th align="left"><p><inline-formula id="IEq160"><alternatives><mml:math id="IEq160_Math"><mml:mrow><mml:mo>-</mml:mo><mml:mspace width="0.166667em"/><mml:mi>L</mml:mi></mml:mrow></mml:math><tex-math id="IEq160_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$-\,L$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq160.gif"/></alternatives></inline-formula></p></th><th align="left"><p>AIC</p></th><th align="left"><p>KS (<italic>p</italic>)</p></th><th align="left"><p><inline-formula id="IEq161"><alternatives><mml:math id="IEq161_Math"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math><tex-math id="IEq161_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${W}^{2}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq161.gif"/></alternatives></inline-formula></p></th></tr></thead><tbody><tr><td align="left"><p>TWE</p></td><td char="." align="char"><p>117.95</p></td><td char="." align="char"><p>239.89</p></td><td char="(" align="char"><p>0.13 (0.12)</p></td><td char="." align="char"><p>0.25</p></td></tr><tr><td align="left"><p>WE</p></td><td char="." align="char"><p>120.65</p></td><td char="." align="char"><p>245.29</p></td><td char="(" align="char"><p>0.13 (0.13)</p></td><td char="." align="char"><p>0.33</p></td></tr><tr><td align="left"><p><inline-formula id="IEq162"><alternatives><mml:math id="IEq162_Math"><mml:mi mathvariant="script">WL</mml:mi></mml:math><tex-math id="IEq162_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$${{\mathcal {WL}}}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq162.gif"/></alternatives></inline-formula></p></td><td char="." align="char"><p>119.71</p></td><td char="." align="char"><p>241.42</p></td><td char="(" align="char"><p>0.17 (0.02)</p></td><td char="." align="char"><p>0.34</p></td></tr></tbody></table></figure></p>
<p>Plots of the fitted densities are shown in Fig. 
<a href="#Fig4"><sup>4</sup></a>
. Left panel of this figure represents the circular data plot, rose diagram and fitted pdf of the TWE, WE and 
<span id="IEq163"><mml:math id="IEq163_Math"><mml:mi mathvariant="script">WL</mml:mi></mml:math><tex-math id="IEq163_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 distributions. The dashed arrow points out the sample mean resultant vector with values 
<span id="IEq164"><mml:math id="IEq164_Math"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1.12</mml:mn><mml:mspace width="3.33333pt"/><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>∼</mml:mo><mml:mn>64</mml:mn><mml:mo>.</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mo>∘</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq164_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$m_{1}=1.12~(\sim 64.2^{\circ })$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq164.gif"/></span>
 and resultant length 
<span id="IEq165"><mml:math id="IEq165_Math"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.4971</mml:mn></mml:mrow></mml:math><tex-math id="IEq165_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$r_{1}=0.4971$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40096_2018_268_Article_IEq165.gif"/></span>
, and the solid arrow points out the mean direction vector and the resultant length of the fitted TWE distribution, which their values are given in Table 
<a href="#Tab2"><sup>2</sup></a>
.
<figure id="Fig4"><h3>Fig. 4</h3>
<figcaption><p>Plots for turtle data. Circular data plot, fitted circular pdf and rose diagram (left), linear histogram and fitted pdf (center), empirical cdf and fitted cdf (right)</p></figcaption>
<img src="40096_2018_268_Fig4_HTML.png" /></figure></p>
<p>According to Table 
<a href="#Tab3"><sup>3</sup></a>
, the TWE distribution has the smallest negative log-likelihood value, AIC and Watson statistics than the others. Thus, we can clearly say that the TWE distribution gives better fit than 
<span id="IEq166"><mml:math id="IEq166_Math"><mml:mi mathvariant="script">WL</mml:mi></mml:math><tex-math id="IEq166_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 distribution and WE distribution. Furthermore, according to the results of the KS test given in Table 
<a href="#Tab3"><sup>3</sup></a>
, the goodness of fit of the 
<span id="IEq167"><mml:math id="IEq167_Math"><mml:mi mathvariant="script">WL</mml:mi></mml:math><tex-math id="IEq167_TeX"><![CDATA[\documentclass[12pt]{minimal}
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 distribution cannot be accepted at a significance level of 0.05.</p></section>
<section><h2>Conclusion</h2>
<p>In this article, we have introduced a new transmuted wrapped distribution named TWE distribution, for modeling the circular data. To the best of our knowledge, the transmutation of a circular distribution is a new attempt to obtain more flexible circular distribution. In the paper, the pdf and the cdf of the introduced distribution are derived and their behaviors are illustrated. Rényi and Shannon entropies of the distribution are obtained in an open form. Furthermore, explicit forms of the basic characteristics of the introduced distribution such as mean, trigonometric moments, characteristic function, quantile function and others are obtained. To estimate the unknown parameters of the introduced distribution, the maximum likelihood, the least squares and the weighted least squares estimators are obtained. By a conducted Monte Carlo simulation study, the efficiencies of these estimators are comparatively illustrated. The results of the Monte Carlo simulation show that when the sample size increases, both Bias and MSE values decrease for all estimation methods. Finally, we apply the introduced distribution to a real-life dataset named turtles dataset. Using the log-likelihood, AIC and Watson’s statistic criteria, the modeling performance of the introduced distribution is compared with wrapped exponential distribution and wrapped Lindley distribution. According to the obtained results, we can say that the TWE distribution is a better model to the turtle data than WE and 
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 distributions.</p></section><hr/><h2>Publisher’s Note</h2>
<p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p>
<hr/><h2>References</h2>
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