A Bayesian-like estimator of the process capability index Cpmk

Pearn et al. (1992) proposed the capability index Cpmk, and investigated the statistical properties of its natural estimator \(\) for stable normal processes with constant mean μ. Chen and Hsu (1995) showed that under general conditions the asymptotic distribution of \(\) is normal if μ≠m, and is a linear combination of the normal and the folded-normal distributions if μ=m, where m is the mid-point between the upper and the lower specification limits. In this paper, we consider a new estimator \(\) for stable processes under a different (more realistic) condition on process mean, namely, P (μ≥m)=p, 0≤p≤1. We obtain the exact distribution, the expected value, and the variance of \(\) under normality assumption. We show that for P (μ≥m)=0, or 1, the new estimator \(\) is the MLE of Cpmk, which is asymptotically efficient. In addition, we show that under general conditions \(\) is consistent and is asymptotically unbiased. We also show that the asymptotic distribution of \(\) is a mixture of two normal distributions.