The Numerical Evaluation of the Maximum-Likelihood Estimate of a Subset of Mixture Proportions

In this note, we give necessary and sufficient conditions for a maximum-likelihood estimate of a subset of the proportions in a mixture of specified distributions. From these conditions, we derive likelihood equations satisfied by the maximum-likelihood estimate and discuss a successive-approximations procedure suggested by these equations for numerically evaluating the maximum-likelihood estimate. It is shown that, with probability 1 for large samples, this procedure converges locally to the maximum- likelihood estimate whenever a certain step-size lies between 0 and 2. Furthermore, optimal rates of local convergence are obtained for a step-size which is bounded below by a number between 1 and 2. 1. Introduction. Let x be an n-dimensional random variable whose density function is a convex combination of density functions po, P1,i.., Pm on Ri. In parti- cular, suppose that the density function of x is p(x, do), a member of the parametric family of density functions m p (x, a)= aipi (x) + (1 - j)po(x) i=l The results given here generalize those of (2), in which a restricted iterative procedure is considered in the special case /3 = 1. We also remark that our results apply to the problem of numerically evaluating a maximum-likelihood estimate of a