Semiparametric Estimation of Symmetric Mixture Models with Monotone and Log-Concave Densities

In this article, we revisit the problem of fitting a mixture model under the assumption that the mixture components are symmetric and log-concave. To this end, we first study the nonparametric maximum likelihood estimation (NPMLE) of a monotone log-concave probability density. By following the arguments of Rufibach (2006), we show that the NPMLE is uniformly consistent with respect to the supremum norm on compact subsets of the interior of the support. To fit the mixture model, we propose a semiparametric EM (SEM) algorithm, which can be adapted to other semiparametric mixture models. In our numerical experiments, we compare our algorithm to that of Balabdaoui and Doss (2014) and other mixture models both on simulated and real-world datasets.