On the Distribution of the Estimated Process Capability Index C pmk

Pearn et al. (Kotz and Johnson, Journal of Quality Technology 24: 216–233, 1992) proposed the process capability index Cpmk , and investigated the statistical properties of the natural estimator for stable normal processes with constant mean μ. Chen and Hsu (Communications in Statistics: Theory and Methods 24: 1279–1291, 1995) showed that under general conditions the asymptotic distribution of the natural estimator 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. Write (Theory & Methods 27: 1781–1789; 1998) derived the probability density function of the natural estimator under normality assumption. In this paper, we consider a new estimator for stable processes under a more general and realistic condition in which the information P(μ ≥ m) = p, 0 ≤ p ≤ 1, is given. We investigate the statistical properties of the new estimator. We obtain the exact distribution, the cumulative distribution function, and the probability density function under normality assumption. We also calculate the expected value and the mean square error of the new estimator for some commonly used parameters.