Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations, II: The Behrens-Fisher Problem

Let $\mu$ be a $p$-dimensional vector, and let $\Sigma_1$ and $\Sigma_2$ be $p \times p$ positive definite covariance matrices. On being given random samples of sizes $N_1$ and $N_2$ from independent multivariate normal populations $N_p(\mu,\Sigma_1)$ and $N_p(\mu,\Sigma_2)$, respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters $\mu$, $\Sigma_1$, and $\Sigma_2$. We shall prove that for $N_1, N_2 > p$ 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.