Multivariate normal distributions, Fisher information and matrix inequalities

Abstract

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A -1 in the class of positive definite Hermitian matrices. Similarly, we prove that A 11 < A -111, where A 11 is the principal submatrix of A and A 11 is the corresponding submatrix of A -1. These results in turn lead to statistical proofs that the the matrix function A -1 is convex in the class of positive definite Hermitian matrices and that A 2 is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A < B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.