Visualizing when prior information can reduce the necessary size of a reference sample when estimating in‐control parameters for Shewhart and CUSUM charts for a normal process

We consider the construction of Shewhart and cumulative sum (CUSUM) charts for a normal process when the in‐control mean and standard deviation must be estimated from a reference sample. Unless the reference sample size is extremely large, substituting the unknown mean and standard deviation with the sample estimates from the reference sample will introduce alarming variability in the conditional (on the reference sample estimates) in‐control average run length. In most applications, some prior information for the mean and standard deviation is available. We investigate how effective Bayes estimators can be in terms of reducing the necessary size of the reference sample. We consider both uniform and conjugate priors, and we show how to construct a graphic that depicts the relationship between the precision in the prior and the relative error in the conditional in‐control average run length.