Finite sample performance of an estimator of process capability index C_pm for the autocorrelated data

Abstract

Process capability index C_pm, sometimes called Taguchi index, is a widely used measure to assess the ability of a process to cluster around the target. In calculating C_pm, it is assumed that the process data are independent, however, in many processes this assumption is violated. Because the exact distribution of estimator ${\tilde{C}}_{pm}$ of C_pm for autocorrelated data is unknown, in this article, we derive explicit expressions for the moments of ${\tilde{C}}_{pm}$ and discuss its statistical properties in terms of these moments. The study shows that the estimator ${\tilde{C}}_{pm}$ is, in general, positively biased. Further, the exact values of mean and standard deviations of ${\tilde{C}}_{pm}$ are compared with the corresponding approximated values provided by earlier studies and it is found that the earlier studies overemphasize the effect of autocorrelation on ${\tilde{C}}_{pm},$ especially for smaller sample sizes, used for calculating ${\tilde{C}}_{pm}.$

Keywords:

• AR (1) model
• Autocorrelation
• Moments
• Process capability index
• Taguchi index

Acknowledgments

The authors would like to thank two anonymous reviewers and the Editor for their helpful and constructive comments that have significantly improved the manuscript.