Bootstrap confidence intervals of CNpk for exponentiated Fréchet distribution

  • Srinivasa Rao GaddeEmail author
  • K. Rosaiah
  • Sridhar Babu Mothukuri
Original Research


Confidence intervals for process capability index using bootstrap method (Chen and Pearn, Qual Reliab Eng Int 13(6), 355–360, 1997) are constructed through simulation assuming that the underlying distribution is exponentiated Fréchet distribution (EFD). Parameters are estimated by Maximum likelihood (ML) method. Also obtain the estimated coverage probabilities and average widths of the bootstrap confidence intervals through Monte Carlo simulation. Illustrate the process capability indices for EFD using some numerical examples.


Exponentiated Fréchet distribution Process capability index Bootstrap confidence interval Maximum likelihood estimation Monte Carlo simulation 


  1. Al-Nassar AD, Al-Omari AI (2013) Acceptance sampling plan based on truncated life tests for exponentiated Fréchet distribution. J Stat 25(2):107–119Google Scholar
  2. Chen KS, Pearn WL (1997) An application of non-normal process capability indices. Qual Reliab Eng Int 13(6):355–360Google Scholar
  3. Chen KS, Huang ML, Hung YH (2008) Process capability analysis chart with the application of Cpm. Int J Prod Res 46(16):4483–4499zbMATHGoogle Scholar
  4. Choi IS, Bai DS (1996) Process capability indices for skewed distributions. In: Proceedings of 20th international conference on computer and industrial engineering, Kyongju, Korea, pp 1211–1214Google Scholar
  5. Clements JA (1989) Process Capability calculations for non-normal distribution. Qual Progr 24(9):95–100Google Scholar
  6. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman and Hall, New YorkzbMATHGoogle Scholar
  7. Folks JL, Chhikara RS (1978) The inverse Gaussian distribution and its statistical application—a review. J R Stat Soc Ser B 40(3):263–289MathSciNetzbMATHGoogle Scholar
  8. Franklin AF, Wasserman GS (1991) Bootstrap confidence interval estimation of C pk: an introduction. Commun Stat Simul Comput 20(1):231–242zbMATHGoogle Scholar
  9. Johnson N, Kotz S, Pearn WL (1994) Flexible process capability indices. Pak J Stat 10(1):23–31MathSciNetzbMATHGoogle Scholar
  10. Juran JM, Gryna FM, Binghan RS (1974) Quality control handbook. McGraw-Hill, New YorkGoogle Scholar
  11. Kane VE (1986) Process capability indices. J Qual Technol 18(1):41–52Google Scholar
  12. Kashif M, Aslam M, Rao GS, Ali HM, Jun C-H (2017a) Bootstrap confidence intervals of the modified process capability index for weibull distribution. Arab J Sci Eng 42:4565–4573Google Scholar
  13. Kashif M, Aslam M, Ali HM, Jun C-H, Khan MI (2017b) Evaluation of modified non-normal process capability index and its bootstrap confidence intervals. IEEE Access 5:12135–12142Google Scholar
  14. Kocherlakota S, Kocherlakota K (1994) Confidence intervals for the process capability ratio based on robust estimators. Commun Stat Theory Methods 23(1):257–276MathSciNetzbMATHGoogle Scholar
  15. Kocherlakota S, Kocherlakota K, Kirmani SNUA (1992) Process capability indices under non-normality, International. J Math Stat Sci 1:175–209zbMATHGoogle Scholar
  16. Nadarajah S, Kotz S (2006) The exponentiated type distributions. Acta Applicandae Mathematicae 92(2):97–111MathSciNetzbMATHGoogle Scholar
  17. Pearn WL, Chen KS (1995) Estimating process capability indices for non-normal Pearson populations. Qual Reliab Eng Int 11(5):386–388Google Scholar
  18. Peng C (2010a) Parametric lower confidence limits of quantile-based process capability indices. Qual Technol Quant Manage 7(3):199–214Google Scholar
  19. Peng C (2010b) Estimating and testing quantile-based process capability indices for processes with skewed distributions. J Data Sci 8(2):253–268Google Scholar
  20. Perakis M (2010) Estimation of differences between process capability indices C pm or C pmk for two processes. J Stat Comput Simul 80(3):315–334MathSciNetzbMATHGoogle Scholar
  21. Rao GS, Aslam M, Kantam RRL (2016a) Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions. J Stat Comput Simul 86(5):862–873MathSciNetGoogle Scholar
  22. Rao GS, Rosaiah K, Babu MS (2016b) Estimation of stress-strength reliability from exponentiated fréchet distribution. Int J Adv Manufac Technol 86(9–12):3041–3049Google Scholar
  23. Sanku D, Saha M, Sudhansu SM, Jun C-H (2018) Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions. Commun Stat Simul Comput 47(1):249–262zbMATHGoogle Scholar
  24. Sundaraiyer VH (1996) Estimation of a process capability index for Inverse Gaussian distribution. Commun Stat Theory Methods 25(10):2381–2393MathSciNetzbMATHGoogle Scholar
  25. Vännman KA (1995) Unified approach to process capability indices. Stat Sinica 5(2):805–820zbMATHGoogle Scholar
  26. Wararit P, Somchit W (2009) Bootstrap confidence intervals of the difference between two process capability indices for half logistic distribution. Pak J Stat Oper Res 8(4):879–894MathSciNetGoogle Scholar
  27. Wu SF, Liang MC (2010) A note on the asymptotic distribution of the process capability index Cpmk. J Stat Comput Simul 80(2):227–235MathSciNetGoogle Scholar

Copyright information

© Society for Reliability and Safety (SRESA) 2018

Authors and Affiliations

  1. 1.Department of StatisticsThe University of DodomaDodomaTanzania
  2. 2.Department of StatisticsAcharya Nagarjuna UniversityGunturIndia

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