The Non-Central Wishart Distribution and Certain Problems of Multivariate Statistics

Abstract

The non-central Wishart distribution is the joint distribution of the sums of squares and cross-products of the deviations from the sample means when the observations arise from a set of normal multivariate populations with constant covariance matrix but expected values that vary from observation to observation. The characteristic function for this distribution is obtained from the distribution of the observations (Theorem 1). By using the characteristic functions it is shown that the convolution of several non-central Wishart distributions is another non-central Wishart distribution (Theorem 2). A simple integral representation of the distribution in the general case is given (Theorem 3). The integrand is a function of the roots of a determinantal equation involving the matrix of sums of squares and cross-products of deviations of observations and the matrix of sums of squares and cross-products of deviations of corresponding expected values. The knowledge of the non-central Wishart distribution is applied to two general problems of multivariate normal statistics. The moments of the generalized variance, which is the determinant of sums of squares and cross-products multiplied by a constant, are given for the cases of the expected values of the variates lying on a line (Theorem 4) and lying on a plane (Theorem 5). The likelihood ratio criterion for testing linear hypotheses can be expressed as the ratio of two determinants or as a symmetric function of the roots of a determinantal equation. In either case there is involved a matrix having a Wishart distribution and another matrix independently distributed such that the sum of these two matrices has a non-central Wishart distribution. When the null hypothesis is not true the moments of this criterion are given in the non-central planar case (Theorem 6).