The asymptotic distribution of the estimated process capability index C_pk

Abstract

Bissell (1990) proposed an estimator C_pk for the process capability index C_pk 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 Ĉz_pk only requires the knowledge of P(μ>= m) = p, where 0 <=p <= 1. The new estimator Ĉ_pk is unbiased, and the variance is smaller than that of Bissell's.

In this paper, we investigated the asymptotic properties of the estimator Ĉ_pk under general conditions. We derived the limiting distribution of Ĉ_pk for arbitrary population assuming the fourth moment exists. The asymptotic distribution provides some insight into the properties of Ĉ_pk which may not be evident from its original definition.

Keywords:

• process capability index
• asymptotic distribution
• process mean
• process standard deviation