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In this paper, the exact distribution of the largest eigenvalue of a singularrandom matrix for multivariate analysis of variance (MANOVA) is discussed. Thekey to developing the distribution theory of eigenvalues of a singular randommatrix is to use heterogeneous hypergeometric functions with two matrixarguments. In this study, we define the singular beta F-matrix and extend thedistributions of a nonsingular beta F -matrix to the singular case. We alsogive the joint density of eigenvalues and the exact distribution of the largesteigenvalue in terms of heterogeneous hypergeometric functions.

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