# Maximum Likelihood Estimation of a Semiparametric Two-component Mixture Model using Log-concave Approximation

## TL;DR

This paper introduces a semiparametric maximum likelihood estimation method for a two-component mixture model with a known component, assuming the unknown component's density is log-concave, with applications in biology and astronomy.

## Contribution

It develops a novel EM algorithm for joint estimation of mixing proportions and unknown distribution under log-concavity assumptions, with proven identifiability and consistency.

## Key findings

- Estimator performs well in simulations
- Method successfully applied to biological data
- Approach outperforms existing estimators in certain scenarios

## Abstract

Motivated by studies in biological sciences to detect differentially expressed genes, a semiparametric two-component mixture model with one known component is being studied in this paper. Assuming the density of the unknown component to be log-concave, which contains a very broad family of densities, we develop a semiparametric maximum likelihood estimator and propose an EM algorithm to compute it. Our new estimation method finds the mixing proportions and the distribution of the unknown component simultaneously. We establish the identifiability of the proposed semiparametric mixture model and prove the existence and consistency of the proposed estimators. We further compare our estimator with several existing estimators through simulation studies and apply our method to two real data sets from biological sciences and astronomy.