The asymptotic distribution of the estimated process capability index (C)over-tilde(pk)

Abstract

Bissell (1990) proposed an estimator <(C)over cap (pk)> for the process capability index C-pk assuming that P(mu greater than or equal to m) = 0, or 1, where mu is the process mean, and m is the midpoint between the upper and lower specification limits. Pearn and Chen (1996) considered a new estimator <(C)over tilde (pk)>, which relaxes Bissell's assumption on the process mean. The evaluation of <(C)over tilde (pk)> only requires the knowledge of P(mu greater than or equal to m) = p, where 0 less than or equal to p less than or equal to 1. The new estimator <(C)over bar (pk)> is unbiased, and the variance is smaller than that of Bissell's. In this paper, we investigated the asymptotic properties of the estimator <(C)over tilde (pk)> under general conditions. We derived the limiting distribution of <(C)over tilde (pk)> for arbitrary population assuming the fourth moment exists. The asymptotic distribution provides some insight into the properties of <(C)over tilde (pk)> which may not be evident from its original definition.