Statistical properties of the |S| multivariate control chart.

Recently there has been a growing interest in multivariate statistical process control in which the simultaneous control of several related quality characteristics is considered. The detection of a shift in the in-control mean vector <(mu)over right arrow>(0) = (mu(0,1),...,mu(0,p))' has already been studied in depth assuming that the p variables to be controlled are distributed as a p-multivariate normal. However, very little attention has been paid to the detection of shifts in the dispersion of the variables, i. e., shifts in the in-control covariance matrix Sigma(0). The use of a chart which consists of plotting values of the generalized variance, \S\, is recommended in the bibliography, but its statistical properties are not well known. In this paper we shall study the distribution of \S\, and we shall show how to obtain a suitable control limit. The power of the chart will be studied and we shall show that it only depends on \Sigma(1)\/\Sigma(0)\, where Sigma(1) is the out-of-control covariance matrix.