International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 1, Pages 147-154
doi:10.1155/S0161171281000100
On the noncentral distribution of the ratio of the extreme roots of wishart matrix
Department of Mathematics & Statistics, Miami University, Oxford 45056, Ohio, USA
Received 6 February 1979
Copyright © 1981 V. B. Waikar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.