{"title":"Confidence intervals of the index $C_{pk}$ for normally distributed quality characteristics using classical and Bayesian methods of estimation","authors":"Mahendra Saha, S. Dey, A. Yadav, Sajid Ali","doi":"10.1214/20-bjps469","DOIUrl":null,"url":null,"abstract":"One of the indicators for evaluating the capability of a process potential and performance in an effective way is the process capability index (PCI). It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the pre-set specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. In this article, we consider a process capability index, $C_{pk}$, suggested by Kane (Journal of Quality Technology 18 (1986) 41–52) which can be used for normal random variables. The objective of this article is three fold: First, we address different methods of estimation of the process capability index $C_{pk}$ from frequentist approaches for the normal distribution. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, Cramer–von-Mises estimators, Anderson–Darling estimators and Right-Tail Anderson–Darling estimators and compare them in terms of their mean squared errors using extensive numerical simulations. Second, we compare three parametric bootstrap confidence intervals (BCIs) namely, standard bootstrap, percentile bootstrap and bias-corrected percentile bootstrap. Third, we consider Bayesian estimation under squared error loss function using normal prior for location parameter and inverse gamma for scale parameter for the considered model. Monte Carlo simulation study has been carried out to compare the performances of the classical BCIs and highest posterior density (HPD) credible intervals of $C_{pk}$ in terms of average widths and coverage probabilities. Finally, two real data sets have been analyzed for illustrative purposes.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/20-bjps469","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 15
Abstract
One of the indicators for evaluating the capability of a process potential and performance in an effective way is the process capability index (PCI). It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the pre-set specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. In this article, we consider a process capability index, $C_{pk}$, suggested by Kane (Journal of Quality Technology 18 (1986) 41–52) which can be used for normal random variables. The objective of this article is three fold: First, we address different methods of estimation of the process capability index $C_{pk}$ from frequentist approaches for the normal distribution. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, Cramer–von-Mises estimators, Anderson–Darling estimators and Right-Tail Anderson–Darling estimators and compare them in terms of their mean squared errors using extensive numerical simulations. Second, we compare three parametric bootstrap confidence intervals (BCIs) namely, standard bootstrap, percentile bootstrap and bias-corrected percentile bootstrap. Third, we consider Bayesian estimation under squared error loss function using normal prior for location parameter and inverse gamma for scale parameter for the considered model. Monte Carlo simulation study has been carried out to compare the performances of the classical BCIs and highest posterior density (HPD) credible intervals of $C_{pk}$ in terms of average widths and coverage probabilities. Finally, two real data sets have been analyzed for illustrative purposes.
期刊介绍:
The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes.
More specifically, the following types of contributions will be considered:
(i) Original articles dealing with methodological developments, comparison of competing techniques or their computational aspects.
(ii) Original articles developing theoretical results.
(iii) Articles that contain novel applications of existing methodologies to practical problems. For these papers the focus is in the importance and originality of the applied problem, as well as, applications of the best available methodologies to solve it.
(iv) Survey articles containing a thorough coverage of topics of broad interest to probability and statistics. The journal will occasionally publish book reviews, invited papers and essays on the teaching of statistics.