The performance of the hypergeometric np chart with estimated parameter

Although it is well known that the performance of attribute control charts decreases significantly when the assumption of known process parameters is invalid, this assumption is prevalent in the pertinent literature. However, in most practical applications, the process parameters have to be estimated from a finite in-control Phase I sample, and therefore the performance of attribute control charts should be evaluated from the perspective of estimated process parameters. In this paper, we compare the run length properties of the hypergeometric np chart in both the known and estimated parameter cases. In particular, we investigate the required number of Phase I samples and new specific chart parameters that allow the hypergeometric np chart with estimated parameter p to have approximately the same in-control performance as in the known parameters case. Moreover, we perform a comprehensive in-control and out-of-control comparison of the hypergeometric np chart with its binomial counterpart. In order to achieve these objectives, we also present a new approach to effectively compute the probability distribution of the sum of independent and identically hypergeometric-distributed random variables. The proposed approximation reduces the computational effort to a few seconds while keeping a remarkable high accuracy with only negligible deviations compared to the exact distribution obtained via convolution.