Classical and Bayesian inference ofC_pyfor generalized Lindley distributed quality characteristic

Abstract

The process capability index (PCI) is a quality control–related statistic mostly used in the manufacturing industry, which is used to assess the capability of some monitored process. It is of great significance to quality control engineers as it quantifies the relation between the actual performance of the process and the preset specifications of the product. Most of the traditional PCIs performed well when process follows the normal behaviour. However, using these traditional indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, Cpy, suggested by Maiti et al, which can be used for normal as well as non‐normal random variables. This article addresses the different methods of estimation of the PCI Cpy from both frequentist and Bayesian view points of generalized Lindley distribution suggested by Nadarajah et al. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, least square and weighted least square estimators, and maximum product of spacings estimators. Next, we consider Bayes estimation under squared error loss function using gamma priors for both shape and scale parameters for the considered model. We use Tierney and Kadane's method as well as Markov Chain Monte Carlo procedure to compute approximate Bayes estimates. Besides, two parametric bootstrap confidence intervals using frequentist approaches are provided to compare with highest posterior density credible intervals. Furthermore, Monte Carlo simulation study has been carried out to compare the performances of the classical and the Bayes estimates of Cpy in terms of mean squared errors along with the average width and coverage probabilities. Finally, two real data sets have been analysed for illustrative purposes.

Inference of Process Capability Index Cpy for 3-Burr-XII Distribution Based on Progressive Type-II Censoring

“…So, samples of c can be easily generated using any gamma-generating routine. In addition, since the conditional posteriors of α and θ in (19) and (20), respectively, do not give standard forms, and therefore Gibbs sampling is not a straightforward choice, and it is appropriate to use the Metropolis-Hastings sampler to implement MCMC technique, see Metropolis et al [31]. Because of these conditional distributions in (19) and (20), the following is a hybrid algorithm with Gibbs sampling steps to update parameter c and Metropolis-Hastings sampler steps to update α and θ.…”

“…Ali and Riaz [18] discussed the generalized PCIs from the Bayesian view point under symmetric and asymmetric loss functions for the simple and mixture of generalized lifetime models. Saha et al [19] studied the classical and Bayesian inference of the index C py for generalized Lindley distributed quality characteristic. e rest of this paper is organized as follows.…”

Objective Bayesian inference for the capability index of the Gamma distribution

“…For nonnormally distributed data, a quantile approach to measure the parametric estimation of process capability indices was developed by Clements 12 . Hence, more researchers focused on the study of the process capability analysis of the nonnormal data; for more details, refer Kashif et al, 13 Panichkitkosolkul, 14 Leiva et al., 15 Senvar and Sennaroglue, 16 Saha et al., 17,18 Meng et al., 19 Mahmoud et al, 20 and Wu 21 …”

Parametric inference of the loss based index Cpm for normal distribution

“…Since then, several researchers have developed numerous techniques for constructing CIs. In this regard, readers may refer to Peng 16,17 ; Leiva et al 18 ; Pearn et al 19,20 ; Kashif et al 21,22 ; Weber et al 23 ; Pina-Monarrez et al 24 ; Rao et al 25 ; Dey et al 26 ; Dey and Saha 27,28 ; Park et al 29 ; Saha et al [30][31][32] , Alomani et al 33 and the references cited therein.…”