Skewed zero-bound distributions and process capability indices for upper specifications

Abstract

A common practical situation in process capability analysis, which is not well developed theoretically, is when the quality characteristic of interest has a skewed distribution with a long tail towards relatively large values and an upper specification limit only exists. In such situations, it is not uncommon that the smallest possible value of the characteristic is 0 and this is also the best value to obtain. Hence a target value 0 is assumed to exist. We investigate a new class of process capability indices for this situation. Two estimators of the proposed index are studied and the asymptotic distributions of these estimators are derived. Furthermore, we suggest a decision procedure useful when drawing conclusions about the capability at a given significance level, based on the estimated indices and their asymptotic distributions. A simulation study is also performed, assuming that the quality characteristic is Weibull-distributed, to investigate the true significance level when the sample size is finite.

Keywords:

• capability index
• skewed distributions
• one-sided specification interval
• upper specification limit
• zero-bound process data
• target value 0
• hypothesis testing
• Weibull distribution

Acknowledgements

The authors are grateful for the financial support provided by the Swedish Research Council (project number 621-2002-3823). In addition, we appreciate the very thorough and thoughtful comments of two anonymous referees that have helped to improve and clarify our presentation.