Summary: The process capability index \(C_{pk}\) has been widely used in the manufacturing industry to provide numerical measures on process performance. Since \(C_{pk}\) is a yield-based index which is independent of the target T, it fails to account for process centering with symmetric tolerances, and presents an even greater problem with asymmetric tolerances. W. L. Pearn and K. S. Chen [J. Appl. Stat. 25, No. 6, 801–810 (1998; Zbl 0945.62127)] considered a new generalized \(C''_{pk}\) which was shown to be superior to other existing generalizations of \(C_{pk}\) for processes with asymmetric tolerances.
We investigate the relation between the fraction nonconforming and the value of \(C''_{pk}\). Furthermore, we derive explicit forms of the cumulative distribution function and the probability density function for the natural estimator \({\widehat C}''_{pk}\), under the assumption of normality. We also develop a decision making rule based on the natural estimator \({\widehat C}''_{pk}\), which can be used to test whether the process is capable or not.