Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Abstract.

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index .