Counting and locating the solutions of polynomial systems of maximum likelihood equations, II: The Behrens-fisher problem

Let μ be a p-dimensional vector, and let Σ₁ and Σ₂ be p × p positive definite covariance matrices. On being given random samples of sizes N₁ and N₂ from independent multivariate normal populations N_p(μ, Σ₁) and N_p(μ, Σ₂), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters μ, Σ₁, and Σ₂. We prove that for N₁, N₂ > p there are, almost surely, exactly 2p + 1 complex solutions of the likelihood equations. For the case in which p = 2, we utilize Monte Carlo simulation to estimate the relative frequency with which a typical Behrens-Fisher problem has multiple real solutions; we find that multiple real solutions occur infrequently.