Comparison of Bootstrap Confidence Interval Methods for Process Capability Indices Cp & Cpk
Abstract
In this paper, some bootstrap confidence interval methods, namely standard bootstrap (SB), percentile bootstrap (PB), percentile t–bootstrap (PTB), bias corrected percentile bootstrap (BCPB), bias corrected and accelerated bootstrap (BCa) are considered for obtaining confidence intervals of commonly used process capability indices Cp and Cpk under normal and non-normal process distributions. The performance of these approaches for the capability indices Cp and Cpk is compared in terms of coverage probability and expected length using simulation. The results of simulation showed that among these confidence intervals the coverage probability and expected length of bias corrected percentile bootstrap (BCPB) approach is best.
References
1. A. Luceño; A Process Capability Index with Reliable Confidence Intervals. Communications in Statistics- Simulation and Computation, 25(1), 235-245 (1996).
2. A. Parchami and M. Mashinchi; Interval Estimation of an Extended Capability Index with Application in Educational Systems. Turkish Journal of Fuzzy Systems, 2(2), 64-76 (2011).
3. B. Efron; Bootstrap Methods: Another Look at Jackknife. The Annals of Statistics, 7(1), 1-26 (1979).
4. B. Efron; The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, (1982).
5. B. Efron and R. Tibshrirani; Bootstrap Methods for Standard Errors, Confidence Intervals and Other Measures of Statistical Accuracy. Statistical Science, 1(1), 54-75(1986).
6. B. Efron; Better Bootstrap Confidence Intervals. Journal of American Statistical Association, 82(397), 171-200 (1987).
7. B. J. Yum and K. W. Kim; A Bibliography of the Literature on Process Capability Indices: 2000-2009, Quality and Reliability Engineering International, 27(3), 251-268 (2011).
8. B. J. Yum; A Bibliography of the Literature on Process Capability Indices (PCIs): 2010-2021, Part I: Books, Review/Overview Papers, and Univariate PCI-related Paper, Quality and Reliability Engineering International, (2023). Doi: https://doi.org/10.1002/qre.3258
9. C. M. Wang and C. T. Lam; Confidence Limits for Proportion of Conformance. Journal of Quality
Technology, 28(4), 439-445 (1996).
10. C. Park; S. Dey; L. Ouyang; J. H. Byun and M. Leeds; Improved Bootstrap Confidence Intervals for the Process Capability Index Cpk. Communications in Statistics-Simulation and Computation, 49(10), 2583-2603 (2020).
11. G. S. Rao; M. Albassam and M. Aslam; Evaluation of Bootstrap Confidence Intervals using a New Non-Normal Process Capability Index. Symmetry, 11(4), 484 (2019).
12. F. Spiring; B. Leung; S. Cheng and A. Yeung; A Bibliography of Process Capability Papers. Quality and Reliability Engineering International, 19(5), 445-460 (2003).
13. J. Han; J. Cho and C. S. Leem; Bootstrap Confidence Limits for Wright's C5, Communications in Statistics-Theory and Methods, 29(3), 485-505 (2000).
14.J. P. Chen and L. I. Tong; Bootstrap Confidence Interval of the Difference between Two Process Capability Indices. International Journal of Advanced Manufacturing Technology, 21(4), 249-256 (2003). 15. K. Rezaie; M. R. Taghizadeh and B. Ostadi; A Practical Implementation of the Process Capability Indices. Journal of Applied Sciences, 6(5), 1182-1185 (2006).
16.L. A. Franklin and G. Wasserman; Bootstrap Confidence Interval Estimates of Cpk: An introduction. Communications in Statistics-Simulation and Computation, 20(1), 231-242 (1991).
17. L. I. Tong and J. P. Chen; Lower Confidence Limits of Process Capability Indices for Non-Normal Process Distributions. International Journal of Quality & Reliability Management, 15(8), 907-919 (1998).
18. L. I. Tong; H. T. Chen and Y. F. Tai; Constructing BCa Bootstrap Confidence Interval for the difference between Two Non-Normal Process Capability Indices CNpmk. Quality Engineering, 20(2), 209-220 (2008).
19. M. Kashif; M. Aslam; A. H. Al-Marshadi and C. H. Jun; Capability Indices for Non-Normal Distribution using Gini’s Mean Difference as Measure of Variability. IEEE Access, 4, 7322-7330 (2016).
20. M. Kashif; M. Aslam; A. H. Al-Marshadi; C. H. Jun and M. I. Khan; Evaluation of Modified Non-Normal Process Capability Index and Its Bootstrap Confidence Intervals. IEEE Access, 5, 12135-12142 (2017).
21. M. Saha; S. Dey and S. Maiti; Parametric and Non-Parametric Bootstrap Confidence Intervals of CNpk for Exponential Power Distribution. Journal of Industrial and Power Engineering, 35(3), 160-169(2018).
22. M. Saha; S. Kumar and R. Sahu; Comparison of Two Generalized Process Capability Indices by using Bootstrap Confidence Intervals. International Journal of Statistics and Reliability Engineering, 7(1), 187-195 (2020).
23. P. Stoma; M. Stoma; A. Dudziak and J. Caban; Bootstrap Analysis of the Production Processes Capability Assessment. Applied Sciences, 9(24), 5360 (2019).
24. R. Somkhuean and A. Wongkhao; Confidence Intervals for the Common Process Capability Index CP of Normal Distribution. Journal of Statistics Applications and Probability, 11(1), 175-187 (2022).
25.S. Dey; M. Saha; S. S. Maiti and C. H. Jun; Bootstrap Confidence Intervals of Generalized Process Capability Index Cpyk for Lindley and Power Lindley Distributions. Communication in Statistics – Simulation and Computation, 47(1), 249-262 (2018).
26. S. Dey and M. Saha; Bootstrap Confidence Intervals of Generalized Process Capability Index Cpyk using Different Methods of Estimation. Journal of Applied Statistics, 46(10), 1843-1869 (2019).
27. S. Dey and M. Saha; Bootstrap Confidence Intervals of Process Capability Index Spmk using Different Methods of Estimation. Journal of Statistical Computation and Simulation, 90(1), 28-50 (2020).
28. S. Kotz and N. L. Johnson; Process Capability Indices - A Review, 1992-2000 Discussions. Journal of Quality Technology, 34(1), 2–19 (2002).
29. S. Balamurali and M. Kalyanasundaram; Bootstrap Lower Confidence Limits for the Process Capability Indices Cp, Cpk and Cpm. International Journal of Quality and Reliability Management, 19(8/9), 1088-1097 (2002).
30. S. A. Niwitpong and P. Kirdwichai; Confidence Interval for Cp of Non-Normal Distributions using an Adjusted Confidence Interval for Standard Deviation. Thailand Statistician, 6(1), 7-14(2008).
31. S. Ali; M. S. C. Khoo; M. Kashif; Z. Javed and S. Saha; Constructing Bootstrap Confidence Intervals of Process Capability Indices for a Three Parameter Weibull Distribution. Matematika MJIM, 38(1) 21-32 (2022).
32. W. Panichkitkosolkul; Confidence Interval for the Process Capability Index CP based on the Bootstrap –t Confidence Interval for the Standard Deviation. Metodoski zvezki, 11(2), 79-92 (2014).
33.W. Panichkitkosolkul; Confidence Intervals for the Process Capability Index CP Based on Confidence Intervals for Variance under Non-Normality. Malaysian Journal of Mathematical Sciences, 10(1), 101-115 (2016).
34.W. D. Heavlin; Statistical Properties of Capability Indices. Technical Report 320, Technical Library, Advanced Micro Devices, Inc., Sunnyvale, CA. (1988).
35.Y. Nagata and H. Nagahata; Approximation Formulas for the Lower Confidence Limits of Process Capability Indices. Communication in Statistics – Theory and methods, 25(4), 301-314 (1994).
36.Y. M. Chow and D. B. Owen; On the Distributions of the Estimated Process Capability Indices. Communication in Statistics-Theory and Methods, 18(12), 4549-4560 (1989).
37. Y. M. Chou; D. B. Owen; A. Salvador and A. Borrego; Lower Confidence Limits on Process Capability Indices. Journal of Quality Technology, 22(3), 223-229 (1990).
38. Y. Jian-feng; A Bootstrap Confidence Limit for Process Capability Indices. Systems Engineering (2007). doi: https://api.semanticscholar.org/CorpusID:51140412.
Trion Studio
