Classical and the Bayesian estimation of process capability index Cpy: A comparative study
In this study, to estimate the process capability index Cpy when the process follows different distributions (Lindley, Xgamma, and Akash distribution), we have used five methods of estimation, namely, the maximum likelihood method of estimation, least and weighted least squares method of estimation, maximum product of spacings method of estimation and Bayesian method of estimation. The Bayesian estimation is studied for symmetric loss function with the help of the Metropolis-Hastings algorithm method. The confidence intervals for the index Cpy are constructed based on four bootstrap methods and Bayesian methods. We studied the performances of these estimators based on their corresponding MSEs/risks for the point estimates of Cpy, and average widths AW for interval estimates. To assess the accuracy of the various approaches, Monte Carlo simulations are conducted. It is found that the Bayes estimates performed better than the considered classical estimates in terms of their corresponding risks. To illustrate the performance of the proposed methods, two real data sets are analyzed.
Chatterjee S., Qiu P. (2009). Distribution-free cumulative sum control charts using bootstrap-based control limits. The Annals of Applied Statistics, 3(1), 349–369.
Chan, L. K., Cheng, S. W., and Spiring, F. A. (1988). A new measure of process capability: Cpm. Journal of Quality Technology, 20(3), 162–175.
Chen, M. H. and Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.
Cheng, S. W. and Spiring, F. A. (1989). Assessing process capability: a Bayesian approach. IEE Transactions, 21(1), 97–98.
Cheng, R. C. H. and Amin, N. A. K. (1979). Maximum product-of-spacings estimation with applications to the lognormal distribution. Math Report, 79.
Cheng, R. C. H. and Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society: Series B (Methodological), 45(3), 394–403.
Choi, I. S., and Bai, D. S. (1996). Process capability indices for skewed distributions. Proceedings of 20th International Conference on Computer and Industrial Engineering, Kyongju, Korea, 1211–1214.
Dennis, J. E., and Schnabel, R. B. (1983). Numerical methods for unconstrained optimization and non-linear equations. Prentice-Hall, Englewood Cliffs, NJ.
Dey, S., Saha, M., and Kumar, S. (2021). Parametric Confidence Intervals of Spmk for Generalized Exponential Distribution. American Journal of Mathematical and Management Sciences, 1–22.
Franklin, A. F., and Wasserman, G. S. (1991). Bootstrap confidence interval estimation of Cpk: an introduction. Communications in Statistics - Simulation and Computation, 20(1), 231–242.
Ghitany, M. E., Atieh B., and Nadarajah, S. (2008). Lindley distribution and its application. Mathematics and Computers in Simulation, 78, 493–506.
Gunter, B. H. (1989). The use and abuse of Cpk. Quality Progress, 22(3), 108–109.
Hsiang, T. C., and Taguchi, G. (1985). A tutorial on quality control and assurance – the Taguchi methods. ASA Annual Meeting, Las Vegas, Nevada, 188.
Huiming, Z. Y., Jun, Y. and Liya, H. (2007). Bayesian evaluation approach for process capability based on sub samples. IEEE International Conference on Industrial Engineering and Engineering Management, Singapore, 1200–1203.
Juran, J. M. (1974). Juran’s quality control handbook, 3rd ed. McGraw-Hill, New York, USA.
Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18, 41–52.
Kumar S. (2021). Classical and Bayesian Estimation of the Process Capability Index Cpy Based on Lomax Distributed. In Yadav D.K. (Eds.), Advance Research Trends in Statistics and Data Science (pp. 115–131). MKSES Publication. http://doi.org/10.5281/zenodo.4699531.
Kumar, S., and Saha, M. (2020). Estimation of Generalized Process Capability Indices Cpy for Poisson Distribution. Invertis Journal of Management, 12(2), 123–130.
Kumar, S., Dey, S., and Saha, M. (2019). Comparison between two generalized process capability indices for Burr XII distribution using bootstrap confidence intervals. Life Cycle Reliability And Safety Engineering, 8(4), 347–355.
Kumar, S., Yadav, A. S., Dey, S., and Saha, M. (2021). Parametric inference of generalized process capability index Cpyk for the power Lindley distribution. Quality Technology & Quantitative Management, 1–34.
Kundu, D., and Pradhan, B. (2009). Bayesian inference and life testing plans for generalized exponential distribution. Sci. China Ser. A Math. 52 (special volume dedicated to Professor Z. D. Bai), 1373–1388.
Leiva, V., Marchanta, C., Saulob, H., Aslam, M., and Rojasd, F. (2014). Capability indices for Birnbaum–Saunders processes applied to electronic and food industries. Journal of Applied Statistics, 41(9), 1881–1902.
Li C., Mukherjee A., Su Q., Xie M. (2016). Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples. International Journal of Production Research, 54(24), 7259–7273.
Lin, T. Y., Wu, C. W., Chen, J. C., and Chiou, Y. H. (2011). Applying Bayesian approach to assess process capability for asymmetric tolerances based on Cpmk index. Applied mathematical modelling, 35(9), 4473–4489.
Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society, 20, 102-107.
Maiti, S. S., Saha, M. and Nanda, A. K. (2010). On generalizing process capability indices. Journal of Quality Technology and Quantitative Management, 7(3), 279–300.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6): 1087–1092.
Miao, R., Zhang, X., Yang, D., Zhao, Y. and Jiang, Z. (2011). A conjugate Bayesian approach for calculating process capability indices. Expert Systems with Applications, 38(7), 8099–8104.
Ouyang, L. Y., Wu, C. C., and Kuo, H. L. (2002). Bayesian assessment for some process capability indices. International journal of information and management sciences, 13(3), 1–18.
Pearn, W. L., Kotz, S., and Johnson, N. L. (1992). Distributional and inferential properties of process capability indices. Journal of Quality Technology, 24, 216–231.
Pearn, W. L., Tai, Y. T., Hsiao, I. F., and Ao, Y. P. (2014). Approximately unbiased estimator for non-normal process capability index CNpk. Journal of Testing and Evaluation, 42, 1408–1417.
Pearn, W. L., Wu, C. C. and Wu, C. H. (2015). Estimating process capability index C pk: classical approach versus Bayesian approach. Journal of Statistical Computation and Simulation, 85(10), 2007–2021.
Pearn, W. L., Tai, Y. T., and Wang, H. T. (2016). Estimation of a modified capability index for non-normal distributions. Journal of Testing and Evaluation, 44, 1998–2009.
Perakis, M. and Xekalaki, E. (2002). A process capability index that is based on the proportion of conformance. Journal of Statistical Computation and Simulation, 72(9), 707–718.
Rao, G. S., Aslam, M., and Kantam, R. R. L. (2016). Bootstrap confidence intervals of CNpk for Inverse Rayleigh and Log-logistic distributions. Journal of Statistical Computation and Simulation, 86(5), 862–873.
Ranneby, B. (1984). The maximum spacing method. an estimation method related to the maximum likelihood Method. Scandinavian Journal of Statistics, 11(2), 93–112.
Saxena, S. and Singh, H. P. (2006). A Bayesian estimator of process capability index. Journal of Statistics and Management Systems, 9(2), 269–283.
Seifi, S. and Nezhad, M. S. F. (2017). Variable sampling plan for resubmitted lots based on process capability index and Bayesian approach. The International Journal of Advanced Manufacturing Technology, 88(9-12), 2547–2555.
Saha, M., Kumar, S., Maiti, S. S., and Yadav, A. S. (2018). Asymptotic and bootstrap confidence intervals of generalized process capability index Cpy Cpy for exponentially distributed quality characteristic. Life Cycle Reliability And Safety Engineering, 7(4), 235–243.
Saha, M., Dey, S., Yadav, A. S., and Kumar, S. (2019). Classical and Bayesian inference of C py for generalized Lindley distributed quality characteristic. Quality And Reliability Engineering International, 35(8), 2593–2611.
Saha, M., Kumar, S., Maiti, S. S., Singh Yadav, A., and Dey, S. (2020a). Asymptotic and bootstrap confidence intervals for the process capability index cpy based on Lindley distributed quality characteristic. American Journal Of Mathematical And Management Sciences, 39(1), 75–89.
Saha, M., Kumar, S., and Sahu, R. (2020b). Comparison of two generalized process capability indices by using bootstrap confidence intervals. International Journal of Statistics and Reliability Engineering, 7(1), 187–195.
Shanker, R. (2015): Akash Distribution and Its Applications. International Journal of Probability and Statistics, 4(3): 65–75.
Sen, S., Maiti, S. S., and Chandra, N. (2016). The xgamma distribution: statistical properties and application. Journal of Modern Applied Statistical Methods, 15(1), 38.
Shiau, J. J. H., Chiang, C. T. and Hung, H. N. (1999a). A Bayesian procedure for process capability assessment. Quality and Reliability Engineering International, 15(5), 369–378.
Shiau, J. J. H., Hung, H. N. and Chiang, C. T. (1999b). A note on Bayesian estimation of process capability indices. Statistics and Probability Letters, 45(3), 215–224.
Smithson, M. (2001). Correct confidence intervals for various regression effect sizes and parameters: the importance of non-central distributions in computing intervals. Educational and Psychological Measurement, 61, 605–632.
Smith, A. F. and Roberts, G. O. (1993). Bayesian computation via the gibbs sampler and related markov chain monte carlo methods. Journal of the Royal Statistical Society. Series B (Methodological), 55.
Swain, J. J., Venkatraman, S. and Wilson, J. R. (1988). Least-squares estimation of distribution functions in Johnson’s translation system. Journal of Statistical Computation and Simulation, 29(4), 271–297.
Tong, G. and Chen, J. P. (1998). Lower confidence limits of process capability indices for non-normal distributions. Quality Engineering, 9, 305–316.
Zimmer, W. J., Keats, J. B., and Wang, F. K. (1998). The Burr XII Distribution in Reliability Analysis, Journal of Quality Technology, 30, 386–394.