Simultaneous Detection of Shift in Means and Variances

Abstract

Abstract Suppose that X is a p-variate random vector, measured in two populations, with mean vector μ j and covariance matrix ψ j in population j,j = 1, 2. In this article we study normal theory estimation of the parameters under the following constraints. It is assumed that (a) the covariance matrices have p identical eigenvectors, (b) in the coordinate system given by the common eigenvectors, only q < p of the means are different between groups, and (c) only the q eigenvalues associated with the same q common eigenvectors are different between groups. There are two main areas of application of these models: (a) in morphometric studies, particularly if size differences between groups are to be removed from the analysis, and (b) in testing for equality of mean vectors and covariance matrices, especially in situations where the number of variables p is large. For applications of type (b), we suggest a randomization test and compare its performance to the T 2 test and to the likelihood ratio test for equality of both populations.