An EM algorithm for fitting a mixture model with symmetric log-concave densities

Abstract

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 (MLE) of a monotone log-concave probability density. 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 (2018, Inference for a two-component mixture of symmetric distributions under log-concavity. Bernoulli 24 (2):1053–71) and other mixture models both on simulated and real-world datasets.

Keywords:

• EM algorithm
• mixture models
• log-concave density

Acknowledgement

We are grateful to Kaspar Rufibach and Lutz Dümbgen for their feedback on the adaptation of their work in Section 2. We are also grateful to Günther Walther for answering questions of identifiability and for calling our attention to the work of Balabdaoui and Butucea (2014).

Notes

1 The method is based on an active set implementation and is available in the R package logcondens.mode.

2 Following the definition in Rockafellar (2015), a concave function f is said to be proper if $f(x)>-\infty$ for at least one x and $f(x)<+\infty$ for every x. A closed function is a function that maps closed sets to closed sets.

Additional information

Funding

We would also like to thank two anonymous referees for their input. This work was partially supported by a grant from the US Office of Naval Research (N00014-13-1-0257) and a grant from the US National Science Foundation (DMS 1223137).