Inference for a Mixture of Symmetric Distributions under Log-Concavity

In this article, we revisit the problem of estimating the unknown symmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the 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 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 symmetric density is efficiently computed using the \proglang{R} package \pkg{logcondens.mode}.