Pearn and Chen (1996) considered the process capability index Cpk, and investigated the statistical properties of its natural estimator under various process conditions. Their investigation, however, was restricted to processes with symmetric tolerances. Recently, Pearn and Chen (1998) considered a generalization of Cpk, referred to as Cpk•, to cover processes with asymmetric tolerances. They investigated the statistical properties of the natural estimator of Cpk, and obtained the exact formulae for the expected value and variance. In this paper, we consider a new estimator of Cpk•, assuming the knowledge on P(μ ≥ T) = p is available, where 0 ≤ p ≤ 1, which can be obtained from historical information of a stable process. We obtain the exact distribution of the new estimator assuming the process characteristic follows the normal distribution. We show that the new estimator is consistent, asymptotically unbiased, which converges to a mixture of two normal distributions. We also show that by adding suitable correction factors to the new estimator, we may obtain the UMVUE and the MLE of the generalization Cpk•.