The asymptotic distribution of the estimated process capability index Cpk

Abstract

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

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