On the Sampling Distributions of the Estimated Process Loss Indices with Asymmetric Tolerances

Abstract

Pearn et al. (2006a) proposed a new generalization of expected loss index L″ _e to handle processes with both symmetric and asymmetric tolerances. Putting the loss in relative terms, a user needs only to specify the target and the distance from the target at which the product would have zero worth to quantify the performance of a process. The expected loss index L″ _e may be expressed as L″ _e  = L″ _ot  + L″ _pe , which provides an uncontaminated separation between information concerning the process accuracy and the process precision. In order to apply the theory of testing statistical hypothesis to test whether a process is capable or not under normality assumption, in this paper we first derive explicit form for the cumulative distribution function and the probability density function of the natural estimator of the three indices L″ _ot , L″ _pe , and L″ _e . We have proved that the sampling distributions of and may be expressed as the chi-square distribution and the normal distribution, respectively. And the distribution of can be described in terms of a mixture of the chi-square distribution and the normal distribution. Then, we develop a decision-making rule based on the estimated index . Finally, an example of testing L″ _e is also presented for illustrative purpose.

Keywords:

• Asymmetric tolerances
• Decision-making rule
• Process capability indices
• Process loss indices
• Sampling distributions

Mathematics Subject Classification:

• Primary 62E15
• Secondary 62P30

Acknowledgments

The authors are grateful to the anonymous referees, the Associate Editor, and the Editor for their insightful comments and suggestions, which substantially improve the presentation of this article. The research of Dr. Y. C. Chang was supported by the National Science Council of Republic of China (Grant No. NSC 95-2221-E-231-024).