Explicit maximum likelihood estimators for certain patterned covariance matrices

Abstract

Given a random sample of size N from a p-variate normal distribution with mean vector μ and covariance matrix Σ, max-imum likelihood estimation of Σ is considered when both Σ and Σ⁻¹ have linear structure, that is, when and where G₁,…,G_m and H₁,…,H_n are sets of known p×p symmetric linearly independent matrices. Theorems are given relating m and n and the sets G₁,…,G_m and H₁,…,H_n in the general case and when Σ is totally reducible. Explicit maximum likelihood estimators of σ₁,…,σ _m are found when m = n, and several examples, are given.

Keywords:

• normal distribution
• linear structure of covariance matrices and their inverses
• Moore-Penrose inverse
• intraclass structure