A new wrapped exponential distribution

Introduction

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.

Suppose that X is a real-valued random variable and also Gx and gx are the cumulative distribution function (cdf) and the probability density function (pdf) of X , respectively. Then

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.

The rest of this paper is organized as follows: In " ^{TWE distribution} " 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 " ^Inference " section. A series of simulation experiments for comparing the performance of the obtained estimators are performed in " ^{Monte Carlo simulation study} " section. We analyze a real-life dataset from the literature for illustrative purposes in " ^{Application to real data} " section. Finally, the last section of the paper concludes the study.

TWE distribution

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 [ ⁵ ] introduced the wrapped exponential (WE) distribution with following pdf and cdf,

Fig. 1

As it can be seen from Fig.  ¹ , the TWE distribution is a unimodal distribution. When Λ>00$$\end{document}]]> , the mode of the distribution is zero; otherwise, it differs from zero for some values of the λ , see " ^{Modality Behavior} " section. We can say that the distribution has got often the negative skewness (we say anticlockwise skewness). The parameter λ plays an important role in the mean and variance of the TWE distribution as a heritage of its task in the exponential distribution.

Skewness and kurtosis

For a circular model, the p th central cosine and sine moments are α¯p=Ecospθ-μ1 and β¯p=Esinpθ-μ1 , respectively [ ¹² ]. As a measure of asymmetry, skewness coefficient is calculated by γ1=β¯2V-3/2 for a circular distribution. Hence, the skewness coefficient of TWEλ,Λ is obtained as

γ1=-V-3/2λλsin2μ1-2cos2μ12Λ+c+cΛcλ2+4-λΛc+2cos2μ1-λsin2μ1cλ2+1.

Kurtosis of a circular distribution is γ2=α¯2-ρ141-ρ1-2 . Therefore, kurtosis coefficient of TWEλ,Λ is obtained as

γ2=V-2λ22Λ+2c+Λc+2c1-Λ-4Λcλ4+5λ2+4λsin2μ1+cλ1+λ2-3Λ-6λΛcλ4+5λ2+4λcos2μ1-2Λ-c+Λc2+4λ4c22c44λ2+12λ2+12.

Figure  ² represents the contour plots of circular variance ( V ) , skewness γ1 and kurtosis γ2 of TWE distribution.

Fig. 2

In general, for a constant value of Λ , it can be seen from Fig.  ² that when the λ increases, the circular variance decreases. However, this is not true for some negative values of Λ . Similarly, for a constant value of λ , when Λ increases, the circular variance decreases. As in the circular variance, the skewness decreases when λ increases. On the other hand, when λ increases Kurtosis increases.

Hazard rate function

The hazard rate function hr of Θ∼TWEλ,Λ random variable is

hrθ=fθ1-Fθ=λe-θλ2Λ+c-2Λe-θλ+ΛcΛ+c-Λe-θλe-θλ-c-1,

where θ∈0,2π,λ>00$$\end{document}]]> , Λ≤1 and c=e-2πλ-1 . Critical point of the hrθ is

θh=1λln-c2(c+1)(Λ-1)2Λ(c+Λ)+2(c+1)Λ(c+Λ)(c+1)(c+Λ)((c+2)Λ+c).

The hazard rate function has bathtub shape when Λ is in the interval

12-c-c2-4c+1,12-c+c2-4c+1.

Here, considering that the smallest value of c is -1 , hrθ appears to be a bathtub in the positive Λ values providing the above condition. We present Fig.  ³ which plots the hazard rate functions of the TWE5.48,0.25 and WE 5.48 distributions for illustrative purposes.

Fig. 3

Inference

In this section, we consider the statistical inference problem for TWEλ,Λ . To estimate the unknown parameters of TWEλ,Λ , we employ the ML, LS and WLS estimation methods commonly used in the literature.

Monte Carlo simulation study

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 λ=0.5,1.5 and Λ=-0.75,0.75 . For the different sample of sizes n=30,50,100 and 1000, the obtained Bias and mean squared error (MSE) values based on the 1000 times replicated simulations are displayed in Table  ¹ .

As can be clearly seen from Table  ¹ , 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  ¹ , we can say that the ML estimators outperform both the LS and the WLS estimators with smaller MSE values.

Application to real data

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 [ ⁶ ]. We obtain the maximum likelihood estimation of the parameters λ and Λ 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 [ ¹⁵ ], for further computation in wrapped distributions.

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  ² , when the dataset is modeled by the TWE distribution.

This dataset was recently used by Joshi and Jose [ ⁷ ] as an application of the wrapped Lindley WL distribution. In order to make a comparison, maximized log likelihood values (L), Akaike information criterion (AIC), Kolmogorov–Smirnov with p values (KS) and Watson’s U2(W2) statistics values for the TWE, WE and WL distributions are given in Table  ³ .

Plots of the fitted densities are shown in Fig.  ⁴ . Left panel of this figure represents the circular data plot, rose diagram and fitted pdf of the TWE, WE and WL distributions. The dashed arrow points out the sample mean resultant vector with values m1=1.12(∼64.2∘) and resultant length r1=0.4971 , 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  ² .

Fig. 4

According to Table  ³ , 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 WL distribution and WE distribution. Furthermore, according to the results of the KS test given in Table  ³ , the goodness of fit of the WL distribution cannot be accepted at a significance level of 0.05.

Conclusion

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 WL distributions.