On Estimation of Generalized Process Capability Index Cpy for One Parameter Polynomial Exponential Family of Distributions and Its Natural Discrete Version

Estimation of generalized process capability index (GPCI), C-py defined as the ratio of the proportion of specification conformance (or process yield) to the proportion of desired (or natural) conformance, is considered under study. The index covers normal as well as nonnormal and continuous as well as discrete quality characteristics. It can be assessed under either unilateral or bilateral specifications. Its expressions have been derived for one parameter polynomial exponential (OPPE), and natural discrete one parameter polynomial exponential (NDOPPE) distributed quality characteristic. Two estimators, maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU), have been derived. Theoretical comparison of these estimators is made with respect to mean squared error (MSE) through a simulation study since the closed-form expression of the maximum likelihood estimator (MLE) of the distributions' parameter is not available. Three data sets are analyzed for illustration purposes; the estimates are compared in respect of estimated variances as MSEs cannot be computed.