Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance

Let X,Y denote two independent real Gaussian p×m and p×n matrices with m,n≥p, each constituted by zero mean independent, identically distributed columns with common covariance. The Roy’s largest root criterion, used in multivariate analysis of variance (MANOVA), is based on the statistic of the largest eigenvalue, Θ1, of (A+B)−1B, where A=XXT and B=Y YT are independent central Wishart matrices. We derive a new expression and efficient recursive formulas for the exact distribution of Θ1. The expression can be easily calculated even for large parameters, eliminating the need of pre-calculated tables for the application of the Roy’s test.