Open Access
February 2021 Confidence intervals of the index $C_{pk}$ for normally distributed quality characteristics using classical and Bayesian methods of estimation
Mahendra Saha, Sanku Dey, Abhimanyu Singh Yadav, Sajid Ali
Braz. J. Probab. Stat. 35(1): 138-157 (February 2021). DOI: 10.1214/20-BJPS469


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, Cramèr–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.


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Mahendra Saha. Sanku Dey. Abhimanyu Singh Yadav. Sajid Ali. "Confidence intervals of the index $C_{pk}$ for normally distributed quality characteristics using classical and Bayesian methods of estimation." Braz. J. Probab. Stat. 35 (1) 138 - 157, February 2021.


Received: 1 August 2019; Accepted: 1 March 2020; Published: February 2021
First available in Project Euclid: 6 January 2021

MathSciNet: MR4195764
Digital Object Identifier: 10.1214/20-BJPS469

Keywords: bootstrap confidence intervals , different classical methods of estimation , normal distribution , Process capability index , sensitivity analysis

Rights: Copyright © 2021 Brazilian Statistical Association

Vol.35 • No. 1 • February 2021
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