Abstract: Letbe 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.