Nonnegative definite solutions to matrix equations with applications to multivariate test statistics
LetX ?N p, n (M, ?), where the meanM is a matrix of orderp × n and the covariance matrix? is a nnd of orderp n. In this paper we first obtain a version of Cochran’s theorem. Basically, the theorem reduces the problem of verifying Wishartness and independence of matrix quadratic forms inX to solving matrix equations in?. Next, we obtain characterizations of the class of nnd solutions? to those matrix equations. As an application we give a simple description of the class of nnd matrices such that the distributions of common multivariate test statistics are invariant except for a scale factor.
AMS 2000 Subject classification Primary 62H10 - 62E15 - Secondary 15A63
Vaish, A., & Chaganty, NR. (2008). Nonnegative definite solutions to matrix equations with applications to multivariate test statistics. Statistical Papers, 49(1), 87-99. https://doi.org/10.1007/s00362-006-0373-0