Abstract: Let be a -dimensional vector, and let and be positive definite covariance matrices. On being given random samples of sizes and from independent multivariate normal populations and , respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters , , and . We prove that for there are, almost surely, exactly complex solutions of the likelihood equations. For the case in which , 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.
Key words and phrases: Behrens-Fisher problem, Bézout's theorem, maximum likelihood estimation, maximum likelihood degree.