A maximum smoothed likelihood based estimation for two component semiparametric density mixtures with a known component

TL;DR

This paper introduces a novel maximum smoothed likelihood estimation method for a semiparametric mixture model with one known component, without assuming additional structure on the unknown density, and demonstrates its effectiveness through theoretical analysis and empirical tests.

Contribution

It develops a new identifiability condition and an iterative MM algorithm for estimating mixture models with a known component without extra assumptions.

Findings

The algorithm converges to a minimizer of the likelihood functional.

The method performs well in simulations and real data applications.

A new identifiability condition for the model is established.

Abstract

We consider a semiparametric mixture of two univariate density functions where one of them is known while the weight and the other function are unknown. Such mixtures have a history of application to the problem of detecting differentially expressed genes under two or more conditions in microarray data. Until now, some additional knowledge about the unknown component (e.g. the fact that it belongs to a location family) has been assumed. As opposed to this approach, we do not assume any additional structure on the unknown density function. For this mixture model, we derive a new sufficient identifiability condition and pinpoint a specific class of distributions describing the unknown component for which this condition is mostly satisfied. Our approach to estimation of this model is based on an idea of applying a maximum smoothed likelihood to what would otherwise have been an ill-posed problem. We introduce an iterative MM (Majorization-Minimization) algorithm that estimates all of the model parameters. We establish that the algorithm possesses a descent property with respect to a log-likelihood objective functional and prove that the algorithm converges to a minimizer of such an objective functional. Finally, we also illustrate the performance of our algorithm in a simulation study and using a real dataset.