10.1007/s40096-018-0265-1

Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process

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  • Derya Karagöz*1,
  1. Department of Statistics, Hacettepe University, Beytepe, Ankara, TR

Published in Issue 2018-10-04

How to Cite

Karagöz, D. (2018). Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process. Mathematical Sciences, 12(4 (December 2018). https://doi.org/10.1007/s40096-018-0265-1
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Abstract

Abstract This paper aims to modify Shewhart, the weighted variance and skewness correction methods in industrial statistical process control. The robust and asymmetric control limits of range chart are constructed to use in contaminated and skewed distributed process. The way of construction of control limits is simple and corresponds to three methods in which sample range estimator is replaced with the robust interquartile range. These three modified methods are evaluated in terms of their type I risks and average run length by using simulation study. The performance of the proposed range charts is assessed when the Phases I and II data are uncontaminated and contaminated. The Weibull, gamma and lognormal distributions are chosen since they can represent a wide variety of shapes from nearly symmetric to highly skewed.

Keywords

  • Skewed distributions,
  • Shewhart method,
  • Weighted variance method,
  • Skewness correction method,
  • Robust estimator
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References

  1. Abu-Shawiesh (2008) A simple robust control chart based on MAD 4(2) (pp. 102-107) https://doi.org/10.3844/jmssp.2008.102.107
  2. Bai and Choi (1995) X¯documentclass[12pt]{minimal}
  3. usepackage{amsmath}
  4. usepackage{wasysym}
  5. usepackage{amsfonts}
  6. usepackage{amssymb}
  7. usepackage{amsbsy}
  8. usepackage{mathrsfs}
  9. usepackage{upgreek}
  10. setlength{oddsidemargin}{-69pt}
  11. begin{document}$$bar{X}$$end{document} and Rdocumentclass[12pt]{minimal}
  12. usepackage{amsmath}
  13. usepackage{wasysym}
  14. usepackage{amsfonts}
  15. usepackage{amssymb}
  16. usepackage{amsbsy}
  17. usepackage{mathrsfs}
  18. usepackage{upgreek}
  19. setlength{oddsidemargin}{-69pt}
  20. begin{document}$$R$$end{document} control charts for skewed populations (pp. 120-131) https://doi.org/10.1080/00224065.1995.11979575
  21. Chan and Cui (2003) Skewness correction X¯documentclass[12pt]{minimal}
  22. usepackage{amsmath}
  23. usepackage{wasysym}
  24. usepackage{amsfonts}
  25. usepackage{amssymb}
  26. usepackage{amsbsy}
  27. usepackage{mathrsfs}
  28. usepackage{upgreek}
  29. setlength{oddsidemargin}{-69pt}
  30. begin{document}$$bar{X}$$end{document} and Rdocumentclass[12pt]{minimal}
  31. usepackage{amsmath}
  32. usepackage{wasysym}
  33. usepackage{amsfonts}
  34. usepackage{amssymb}
  35. usepackage{amsbsy}
  36. usepackage{mathrsfs}
  37. usepackage{upgreek}
  38. setlength{oddsidemargin}{-69pt}
  39. begin{document}$$R$$end{document} charts for skewed distributions (pp. 1-19) https://doi.org/10.1002/nav.10077
  40. Chang and Bai (2001) Control charts for positively skewed populations with weighted standard deviations (pp. 397-406) https://doi.org/10.1002/qre.427
  41. Choobineh and Branting (1986) A simple approximation for semivariance (pp. 364-370) https://doi.org/10.1016/0377-2217(86)90332-2
  42. Choobineh and Ballard (1987) Control-limits of QC charts for skewed distributions using weighted variance (pp. 473-477) https://doi.org/10.1109/TR.1987.5222442
  43. Davis and Adams (2005) Robust monitoring of contaminated data (pp. 163-174) https://doi.org/10.1080/00224065.2005.11980314
  44. Duclos, E., Pillet, M.: An optimal control chart for non-normal process. In:
  45. 1st
  46. usepackage{amsmath}
  47. usepackage{wasysym}
  48. usepackage{amsfonts}
  49. usepackage{amssymb}
  50. usepackage{amsbsy}
  51. usepackage{mathrsfs}
  52. usepackage{upgreek}
  53. setlength{oddsidemargin}{-69pt}
  54. begin{document}$$1^{st}$$end{document}]]>
  55. IFAC Workshop on Manufacturing Systems: Manufacturing Modeling, Management and Control, Austria 12p (1997)
  56. He and Fung (1999) Method of medians for lifetime data with Weibull models (pp. 1993-2009) https://doi.org/10.1002/(SICI)1097-0258(19990815)18:15<1993::AID-SIM165>3.0.CO;2-H
  57. Khodabin and Ahmadabadi (2010) Some properties of generalized gamma distribution 4(1) (pp. 9-28)
  58. Huang et al. (2017) A control chart for the lognormal standard deviation 15(1) (pp. 1-36) https://doi.org/10.1080/16843703.2017.1304044
  59. Jensen et al. (2006) Effects of parameter estimations on control chart properties: a literature review 38(4) (pp. 349-364) https://doi.org/10.1080/00224065.2006.11918623
  60. Karagöz and Hamurkaroğlu (2012) Control charts for skewed distributions: Weibull, Gamma, and Lognormal 9(2) (pp. 95-106)
  61. Koyuncu and Karagöz (2017) New mean charts for bivariate asymmetric distributions using different ranked set sampling designs 15(5) (pp. 602-621) https://doi.org/10.1080/16843703.2017.1321220
  62. Montgomery (1997) Wiley
  63. Rocke (1989) Robust control charts 31(2) (pp. 173-184) https://doi.org/10.1080/00401706.1989.10488511
  64. Rocke (1992) X^Qdocumentclass[12pt]{minimal}
  65. usepackage{amsmath}
  66. usepackage{wasysym}
  67. usepackage{amsfonts}
  68. usepackage{amssymb}
  69. usepackage{amsbsy}
  70. usepackage{mathrsfs}
  71. usepackage{upgreek}
  72. setlength{oddsidemargin}{-69pt}
  73. begin{document}$$hat{X}_Q$$end{document} and RQdocumentclass[12pt]{minimal}
  74. usepackage{amsmath}
  75. usepackage{wasysym}
  76. usepackage{amsfonts}
  77. usepackage{amssymb}
  78. usepackage{amsbsy}
  79. usepackage{mathrsfs}
  80. usepackage{upgreek}
  81. setlength{oddsidemargin}{-69pt}
  82. begin{document}$$R_Q$$end{document} charts: robust control charts 41(1) (pp. 97-104) https://doi.org/10.2307/2348640
  83. Schoonhoven and Does (2010) The X¯documentclass[12pt]{minimal}
  84. usepackage{amsmath}
  85. usepackage{wasysym}
  86. usepackage{amsfonts}
  87. usepackage{amssymb}
  88. usepackage{amsbsy}
  89. usepackage{mathrsfs}
  90. usepackage{upgreek}
  91. setlength{oddsidemargin}{-69pt}
  92. begin{document}$$bar{X}$$end{document} control chart under non-normality (pp. 167-176)
  93. Schoonhoven et al. (2011) Design and analysis of control charts for standard deviation with estimated parameters (pp. 307-333) https://doi.org/10.1080/00224065.2011.11917867
  94. Schoonhoven and Does (2012) A robust standard deviation control chart 54(1) (pp. 73-82) https://doi.org/10.1080/00401706.2012.648869
  95. Schoonhoven and Does (2013) A robust X¯documentclass[12pt]{minimal}
  96. usepackage{amsmath}
  97. usepackage{wasysym}
  98. usepackage{amsfonts}
  99. usepackage{amssymb}
  100. usepackage{amsbsy}
  101. usepackage{mathrsfs}
  102. usepackage{upgreek}
  103. setlength{oddsidemargin}{-69pt}
  104. begin{document}$$bar{X}$$end{document} control chart (pp. 951-970) https://doi.org/10.1002/qre.1447
  105. Sindhumol et al. (2016) A robust dispersion control chart based on modified trimmed standard deviation 9(1) (pp. 111-121)
  106. Sukparungsee (2013) Asymmetric Tukey’s control chart robust to skew and non-skew process observation 7(8) (pp. 1275-1278)
  107. Tang and Yeh (2016) Approximate confidence intervals for the log-normal standard deviation (pp. 715-725) https://doi.org/10.1002/qre.1786
  108. Tatum (1997) Robust estimation of the process standard deviation for control charts (pp. 127-141) https://doi.org/10.1080/00401706.1997.10485078
  109. Vargas (2003) Robust estimation in multivariate control charts for individual observations (pp. 367-376) https://doi.org/10.1080/00224065.2003.11980234