Phase I and Phase II control charts for the variance and generalized variance

Abstract

By extending the results of Human, Chakraborti and Smit (2010), Phase I control charts are derived for the generalized variance when the mean vector and covariance matrix of multivariate normally distributed data are unknown and estimated from m independent samples, each of size n. In Phase II predictive distributions based on a Bayesian approach are used to construct Shewart-type control limits for the variance and generalized variance. The posterior distribution is obtained by combining the likelihood (the observed data in Phase I) and the uncertainty of the unknown parameters via the prior distribution. By using the posterior distribution the unconditional predictive density functions are derived.