Noncentral Generalized Multivariate Beta Type II Distribution

Abstract

The distribution of the variables that originates from monitoring the variance when the mean encountered a sustained shift is considered — specifically for the case when measurements from each sample are independent and identically distributed normal random variables. It is shown that the solution to this problem involves a sequence of dependent random variables that are constructed from independent noncentral chisquared random variables. This sequence of dependent random variables are the key to understanding the performance of the process used to monitor the variance and are the focus of this article. For simplicity, the marginal (i.e. the univariate and bivariate) distributions and the joint (i.e. the trivariate) distribution of only the first three random variables following a change in the variance is considered. A multivariate generalization is proposed which can be used to calculate the entire run-length (i.e. the waiting time until the first signal) distribution.