Inference for a Two-Component Mixture of Symmetric Distributions under Log-Concavity

TL;DR

This paper studies the estimation of a symmetric mixture model with a log-concave component, demonstrating consistency and convergence rates of the nonparametric MLE under certain conditions.

Contribution

It establishes the consistency and convergence rates of the log-concave MLE for the mixture model with known shift estimators, extending previous work to log-concave densities.

Findings

MLE is consistent in Hellinger distance for the mixed and component densities.

Convergence rate of the MLE in L1 distance is n^{-2/5} when shift estimators are √n-consistent.

Efficient computation of the density is implemented via an R package.

Abstract

In this article, we revisit the problem of estimating the unknown zero-symmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the zero-symmetric distribution has a log-concave density. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown zero-symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are $\sqrt n$-consistent, we establish that these MLE's converge to the truth at the rate $n^{-2/5}$ in the $L_1$ distance. To estimate the shift locations and mixing probability, we use the estimators proposed by \cite{hunteretal2007}. The unknown zero-symmetric density is efficiently computed using the \proglang{R} package \pkg{logcondens.mode}.