Classical Estimation of the Index Spmk and Its Confidence Intervals for Power Lindley Distributed Quality Characteristics

Ghadah Alomani,Sanku Dey,Refah Alotaibi,Mahendra Saha

doi:10.1155/2020/8974349

Abstract

The process capability index (PCI) has been introduced as a tool to aid in the assessment of process performance. Usually, conventional PCIs perform well under normally distributed quality characteristics. However, when these PCIs are employed to evaluate nonnormally distributed process, they often provide inaccurate results. In this article, in order to estimate the PCI Spmk when the process follows power Lindley distribution, first, seven classical methods of estimation, namely, maximum likelihood method of estimation, ordinary and weighted least squares methods of estimation, Cramèr–von Mises method of estimation, maximum product of spacings method of estimation, Anderson–Darling, and right-tail Anderson–Darling methods of estimation, are considered and the performance of these estimation methods based on their mean squared error is compared. Next, three bootstrap confidence intervals (BCIs) of the PCI Spmk, namely, standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap, are considered and compared in terms of their average width, coverage probability, and relative coverage. Besides, a new cost-effective PCI, namely, Spmkc is introduced by incorporating tolerance cost function in the index Spmk. To evaluate the performance of the methods of estimation and BCIs, a simulation study is carried out. Simulation results showed that the maximum likelihood method of estimation performs better than their counterparts in terms of mean squared error, while bias-corrected percentile bootstrap provides smaller confidence length (width) and higher relative coverage than standard bootstrap and percentile bootstrap across sample sizes. Finally, two real data examples are provided to investigate the performance of the proposed procedures.

Highlights

Any process performance can be conveniently measured by process capability indices (PCIs)
To evaluate whether the production process is conforming to predefined specifications and to analyze the quality process and productivity, one can adopt PCIs. e fundamental assumption on process capability analysis was that the process was stable and the process quality characteristic of interest was normally distributed, while in practice, it has been observed that very often manufacturing processes follow nonnormally distributed quality characteristics
To assess the performance of bootstrap confidence intervals (BCIs), we report the estimated average widths, coverage probabilities, and relative coverages of BCIs of the indices Spmk and Spmkc for power Lindley distribution (PLD) using maximum likelihood estimator (MLE), least squares estimator (LSE), weighted least squares estimator (WLSE), Cramer–von Mises estimator (CME), maximum product of spacing estimator (MPSE), Anderson–Darling estimator (ADE), and right-tail Anderson–Darling estimator (RTADE), respectively, in Tables 3–9. e comparisons are based on lower AWs, higher CPs, and higher RCs. e calculated coverage probabilities are compared with the nominal value of 95%

Summary

Introduction

Any process performance can be conveniently measured by process capability indices (PCIs). Ese two PCIs are compared using different methods of estimation and BCIs. our aim is to select the best estimation method among the seven different frequentist methods of the indices Spmk and Spmkc which would be of significance to quality control engineers, where the item/subgroup quality characteristic follows power Lindley distribution (PLD). Is the exact proportion of nonconformity and μ and σ are the process mean and process standard deviation, respectively

Estimation of the Indices Spmk and Spmkc

Bootstrap Confidence Intervals

Simulation Study

Applications

Conclusions