COUNTING AND LOCATING THE SOLUTIONS OF POLYNOMIAL SYSTEMS OF MAXIMUM LIKELIHOOD EQUATIONS, II: THE BEHRENS-FISHER PROBLEM
Max-Louis G. Buot, Serkan Hoşten, Donald St, P. Richards · 2007
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 Np(µ, Σ1) and Np(µ, Σ2), respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters µ, Σ1, and Σ2. It is well-known that the likelihood equations cannot be solved explicitly, and this has led to many different approaches to the Behrens-Fisher problem and with a commensurately large number of publications on the topic. We prove that for N1, N2> p, there are, almost surely, exactly 3 p real or complex solutions of the likelihood equations. We propose a new iterative algorithm for solving the system of likelihood equations. For the case in which p = 2, we utilize Monte Carlo simulation to estimate how frequently a typical Behrens-Fisher is likely to have multiple real solutions; we find that multiple real solutions occur infrequently.