A Bayesian-like estimator of the process capability index Cpmk

Abstract. Pearn et al. (1992) proposed the capability index Cpmk, and inves-tigated the statistical properties of its natural estimator Ĉpmk for stable normal processes with constant mean m. Chen and Hsu (1995) showed that under general conditions the asymptotic distribution of Ĉpmk is normal if m0m, and is a linear combination of the normal and the folded-normal distributions if m m, where m is the mid-point between the upper and the lower specifi-cation limits. In this paper, we consider a new estimator ~Cpmk for stable pro-cesses under a di¤erent (more realistic) condition on process mean, namely, PðmbmÞ p, 0a pa 1. We obtain the exact distribution, the expected value, and the variance of ~Cpmk under normality assumption. We show that for PðmbmÞ 0, or 1, the new estimator ~Cpmk is the MLE of Cpmk, which is asymptotically e‰cient. In addition, we show that under general condi-tions ~Cpmk is consistent and is asymptotically unbiased. We also show that the asymptotic distribution of ~Cpmk is a mixture of two normal distributions