Parametric inference of the loss based index Cpm for normal distribution

Abstract

The process capability index (PCI) has been introduced as a tool to aid in the assessment of process performance and most of the traditional PCIs performed well when process follows the normal behavior. In this article, we consider a PCI,  𝑝𝑚 for normal random variables. The objective of this article is five fold: First, six different methods of estimation of the PCI  𝑝𝑚 are addressed from frequentist approaches and compare them in terms of their mean squared errors using extensive numerical simulations. Second, we compare three bootstrap confidence intervals (BCIs) of the PCI  𝑝𝑚 including standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap. Third, Bayesian estimation is considered under four loss functions (symmetric as well as asymmetric) using normal prior for location parameter and inverse gamma prior for the scale parameter for the considered model. Also, we obtain highest posterior density (HPD) credible intervals of the PCI  𝑝𝑚 . Fourth, a new cost effective PCI, namely,  𝑝𝑚𝑐 is introduced by incorporating tolerance cost function in the index  𝑝𝑚 . Fifth, the power of the test is obtained, and Monte Carlo simulation study has been carried out to compare the performances of the classical BCIs and HPD credible intervals of PCIs  𝑝𝑚 and  𝑝𝑚𝑐 in terms of average widths and coverage probabilities. Finally, two real data sets have been analyzed for illustrative purposes.

Bootstrap confidence intervals of process capability indices Spmk, Spmkc and Cs for Frechet distribution

“…Moreover, several other authors have studied Bayesian estimation of PCIs for lifetime distributions namely log-logistic, Gamma and Weibull distributions. See the work; Dey and Saha, 25 Kargar et al, 41 Saxena and Singh, 39 Ali and Riaz, 42 Pearn et al, 32 Saha et al, 43 Huiming et al, 40 and Saha et al 44 The theory of extreme values shows that the Weibull distribution can be used to model the minimum of large number of independent positive random variables while the Frechet distribution (FD) can be used to model the maximum of a large number of random variables from a certain class of distribution. In FD, when scale parameter β = 1, the distribution is the same as the inverse exponential distribution; when β < 1, it follows the inverse gamma distribution and when β = 2, it is known as inverse Rayleigh distribution.…”

“…Moreover, several other authors have studied Bayesian estimation of PCIs for lifetime distributions namely log‐logistic, Gamma and Weibull distributions. See the work; Dey and Saha, 25 Kargar et al., 41 Saxena and Singh, 39 Ali and Riaz, 42 Pearn et al., 32 Saha et al., 43 Huiming et al., 40 and Saha et al 44 …”

Modified estimation and confidence intervals of an asymmetric loss‐based process capability index Cpm′$\mathcal {C}^{\prime }_{pm}$

“…Usually, for monitoring a process using a PCI, quality control engineers and statisticians employ two estimation techniques, namely, point estimation and confidence intervals (CIs) (see Chan et al 3 ). For more information on CIs, readers may refer to the works of Peng, 14 Leiva et al, 15 Pearn et al, 16 Kashif et al, 17 Weber et al, 18 Rao et al, 19 Dey et al, 20 Wu, 21 , Dey and Saha, 22,23 Saha et al, [24][25][26][27][28][29] Alomani et al, 30 and the references cited therein.…”

Classical and objective Bayesian estimation and confidence intervals of an asymmetric loss based capability index Cpmk′$\mathcal {C}^{\prime }_{pmk}$

“…The point estimator is normally employed for evaluation of a process performance, whereas it is not practically useful in making reasonable decisions due to inherent variability associated with a single estimate. One may thus prefer the use of the CIs as they are capable of providing greater information about the process; see, for example, Peng 8 ; Leiva et al 9 ; Pearn et al 10 ; Kashif et al 11 ; Weber et al 12 ; Rao et al 13 ; Dey et al 14 ; Dey and Saha 15,16 ; Saha et al [17][18][19][20][21][22][23] ; Alomani et al, 24 Kumar et al 25 and many others.…”