Bootstrap confidence intervals of the difference between two generalized process capability indices for inverse Lindley distribution

  • Sanku Dey
  • Mahendra Saha
Original Research


A process capability index (PCI) meant for assessing the capability of the concerned manufacturing process to manufacturer’s products as per specifications pre-set by the product designers or customers. In this article, we utilize bootstrap re-sampling simulation method to construct bootstrap confidence intervals, namely, standard bootstrap (s-boot), percentile bootstrap (p-boot), and bias-corrected percentile bootstrap (BCp-boot) for the difference between two indices (\(C_\mathrm{pyk1}-C_\mathrm{pyk2}\)) through simulation when the underlying distribution is inverse Lindley distribution. Maximum-likelihood method is used to estimate the parameter of the model. The proposed bootstrap confidence intervals can be effectively employed to determine which one of the two processes or manufacturer’s (or supplier’s) has a better process capability. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals of (\(C_\mathrm{pyk1}-C_\mathrm{pyk2}\)). Simulation results showed that the estimated coverage probabilities of the standard bootstrap confidence interval perform better than their counterparts. Finally, a simulated data and a real data are presented to illustrate the bootstrap confidence intervals of the difference between two PCIs.


Generalized process capability index Maximum-likelihood estimate Bootstrap confidence intervals Inverse Lindley distribution 



The authors would like to thank the Referee, Editor-in-Chief, and Associate Editor for careful reading and for comments which greatly improved the paper. Also authors would like to thank the Assistant Librarian, Central University of Rajasthan for his valuable support.


  1. Anis MZ, Tahir M (2016) On some subtle misconceptions about process capability indices. Int J Adv Manuf Technol 87:3019–3029CrossRefGoogle Scholar
  2. Boyles RA (1991) The Taguchi capability index. J Qual Technol 23:17–26CrossRefGoogle Scholar
  3. Chan LK, Cheng SW, Spiring FA (1988) A new measure of process capability: \(C_{pm}\). J Qual Technol 20(3):162–175CrossRefGoogle Scholar
  4. Chen JP, Tong LI (2003) Bootstrap confidence interval of the difference between two process capability indices. Int J Adv Manuf Technol 21:249–256CrossRefGoogle Scholar
  5. Choi IS, Bai DS (1996) Process capability indices for skewed distributions. In: Proceedings of 20th international conference on computer and industrial engineering, Kyongju, pp 1211–1214Google Scholar
  6. Chou YM (1994) Selecting a better supplier by testing process capability indices. Qual Eng 6:427–438CrossRefGoogle Scholar
  7. Dey S, Saha M, Maiti SS, Jun C-H (2017) Bootstrap confidence intervals of generalized process capability index \(C_{pyk}\). Commun Stat Simul Comput.
  8. Efron B (1982) The Jackknife, the bootstrap and other re-sampling plans, SIAM, CBMS-NSF Monograph. 38: SIAM: Philadelphia, PennsylvaniaGoogle Scholar
  9. Franklin AF, Wasserman GS (1991) Bootstrap confidence interval estimation of \(C_{pk}\): an introduction. Commun Stat Simul Comput 20(1):231–242CrossRefzbMATHGoogle Scholar
  10. Gunter BH (1989) The use and abuse of \(C_{pk}\). Qual Progress 22(3):108–109Google Scholar
  11. Ihaka R, Gentleman R (1996) R: a language for data analysis and graphics. J Comput Graph Stat 5:299–314Google Scholar
  12. Juran JM (1974) Juran’s quality control handbook, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  13. Kane VE (1986) Process capability indices. J Qual Technol 18:41–52CrossRefGoogle Scholar
  14. Kanichukattu JK, Luke JA (2013) Comparison between two process capability indices using generalized confidence intervals. Int J Adv Manuf Technol 69:2793–2798CrossRefGoogle Scholar
  15. Kashif M, Aslam M, Rao GS, Al-Marshadi AH, Jun CH (2017) Bootstrap confidence intervals of the modified process capability index for Weibull distribution. Arab J Sci Eng. Google Scholar
  16. Kshif M, Aslam M, Al-Marshadi AH, Jun CH (2016) Capability indices for non-normal distribution using Ginis mean difference as measure of variability. IEEE Access 4:7322–7330CrossRefGoogle Scholar
  17. Maiti SS, Saha M, Nanda AK (2010) On generalizing process capability indices. J Qual Technol Quant Manag 7(3):279–300CrossRefGoogle Scholar
  18. Pearn WL, Kotz S, Johnson NL (1992) Distributional and inferential properties of process capability indices. J Qual Technol 24:216–231CrossRefGoogle Scholar
  19. Pearn WL, Tai YT, Hsiao IF, Ao YP (2014) Approximately unbiased estimator for non-normal process capability index \(C_{Npk}\). J Test Eval 42:1408–1417Google Scholar
  20. Pearn WL, Tai YT, Wang HT (2016) Estimation of a modified capability index for non-normal distributions. J Test Eval 44:1998–2009CrossRefGoogle Scholar
  21. Perakis M (2010) Estimation of differences between process capability indices \(C_{pm}\) or \(C_{pmk}\) for two processes. J Stat Comput Simul 80(3):315–334MathSciNetCrossRefzbMATHGoogle Scholar
  22. Pina-Monarrez MR, Ortiz-Yaez JF, Rodrguez-Borbn MI (2016) Non-normal capability indices for the Weibull and Lognormal distributions. Qual Relaib Eng Int 32:1321–1329CrossRefGoogle Scholar
  23. Sharma VK, Singh SK, Singh U, Agarwal V (2015) The inverse Lindley distribution: a stress–strength reliability model with application to head and neck cancer data. J Ind Product Eng 32(3):162–173CrossRefGoogle Scholar
  24. Tong LI, Chen HT, Tai YF (2008) Constructing BCA bootstrap confidence interval for the difference between two non-normal process capability indices CNpmk. Qual Eng 20:209–220CrossRefGoogle Scholar
  25. Tosasukul J, Budsaba K, Volodin A (2009) Dependent bootstrap confidence intervals for a population mean. Thail Stat 7(1):43–51MathSciNetzbMATHGoogle Scholar
  26. Vannman K (1995) A unified approach to capability indices. Stat Sin 5:805–820zbMATHGoogle Scholar
  27. Weber S, Ressurreio T, Duarte C (2016) Yield prediction with a new generalized process capability index applicable to non-normal data. IEEE Trans Comput Aided Des Int Circ Syst 35(931–942):2016Google Scholar

Copyright information

© Society for Reliability and Safety (SRESA) 2018

Authors and Affiliations

  1. 1.Department of StatisticsSt. Anthony’s CollegeShillongIndia
  2. 2.Department of StatisticsCentral University of RajasthanAjmerIndia

Personalised recommendations