Inference for a two-component mixture of symmetric distributions under log-concavity

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 zerosymmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are ? n-consistent, we establish that these MLE’s converge to the truth at the rate n 2{5 in the L1 distance. To estimate the shift locations and mixing probability, we use the estimators proposed by Hunter et al. (2007). The unknown zero-symmetric density is efficiently computed using the R package logcondens.mode. AMS 2000 subject classifications: Primary 62G07, 62H12; secondary 62G05, 62G20.