The asymptotic distribution of the estimated process capability index C̃_pk

Bissell (1990) proposed an estimator Ĉ′_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 C̃_pk, which relaxes Bissell's assumption on the process mean. The evaluation of C̃_pk only requires the knowledge of P(μ ≥ m) = p, where 0 ≤ p ≤ 1. The new estimator C̃_pk is unbiased, and the variance is smaller than that of Bissell's. In this paper, we investigated the asymptotic properties of the estimator C̃_pk under general conditions. We derived the limiting distribution of C̃_pk for arbitrary population assuming the fourth moment exists. The asymptotic distribution provides some insight into the properties of C̃_pk which may not be evident from its original definition.