Counting and locating the solutions of polynomial systems of maximum likelihood equations, I

It is proved that for N1,N2 > p there are, almost surely, exactly 2p+1 complex solutions of the likelihood equations, and Monte Carlo simulation is utilized to estimate the relative frequency with which a typical Behrens-Fisher problem has multiple real solutions.

Abstract

In statistical inference, mixture models consisting of several component subpopulations are used widely to model data drawn from heterogeneous sources. In this paper, we consider maximum likelihood estimation for mixture models in which the only unknown parameters are the component proportions. By applying the theory of multivariable polynomial equations, we derive bounds for the number of isolated roots of the corresponding system of likelihood equations. If the component densities belong to certain familiar continuous exponential families, including the multivariate normal or gamma distributions, then our upper bound is, almost surely, the exact number of solutions.