On the noncentral distribution of the ratio of the extreme roots of wishart matrix

Abstract

The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p×n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix ∑. Further, let S=XX′ and let 11>…>1p>0 be the characteristic roots of S. Thus S has a noncentral Wishart distribution. In this paper, the exact distribution of fp=1−1p/11 is derived. The density of fp is given in terms of zonal polynomials. These results have applications in nuclear physics also.