The Numerical Evaluation of the Maximum-Likelihood Estimate of a Subset of Mixture Proportions
B. C. Peters, Homer F. Walker · SIAM Journal on Applied Mathematics · 1978
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.