The synthetic [Xbar] chart with estimated parameters

TLDR: The goals of this article are to evaluate (using a Markov chain model) the performances of the synthetic [Xbar] chart when the process parameters are estimated, and suggest guidelines concerning the choice of the number of Phase I samples and to provide new optimal constants, especially dedicated to thenumber of samples used in practice.

Abstract: A synthetic [Xbar] chart consists of an integration of a Shewhart [Xbar] chart and a conforming run length chart. This type of chart has been extensively used to detect a process mean shift under the assumption of known process parameters. However, in practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. The goals of this article are to (i) evaluate (using a Markov chain model) the performances of the synthetic [Xbar] chart when the process parameters are estimated; (ii) compare it with the case where the process parameters are assumed known to demonstrate that these performances are quite different when the number of samples used during Phase I is small; and (iii) suggest guidelines concerning the choice of the number of Phase I samples and to provide new optimal constants, especially dedicated to the number of samples used in practice.