On the noncentral distribution of the ratio of the extreme roots of wishart matrix - V. B. Waikar (International Journal of Mathematics and Mathematical Sciences)
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.