Expressing the largest eigenvalue of a singular beta F-matrix with heterogeneous hypergeometric functions

Abstract

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta [Formula: see text]-matrix and extend the distributions of a nonsingular beta [Formula: see text]-matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.