In this article, we revisit the problem of estimating the unknownzero-symmetric distribution in a two-component location mixture model,considered in previous works, now under the assumption that the zero-symmetricdistribution has a log-concave density. When consistent estimators for theshift locations and mixing probability are used, we show that the nonparametriclog-concave Maximum Likelihood estimator (MLE) of both the mixed density andthat of the unknown zero-symmetric component are consistent in the Hellingerdistance. In case the estimators for the shift locations and mixing probabilityare $\sqrt n$-consistent, we establish that these MLE's converge to the truthat the rate $n^{-2/5}$ in the $L_1$ distance. To estimate the shift locationsand mixing probability, we use the estimators proposed by\cite{hunteretal2007}. The unknown zero-symmetric density is efficientlycomputed using the \proglang{R} package \pkg{logcondens.mode}.
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机译：在本文中，我们假设零对称分布具有对数凹凹密度，在此之前，我们回顾了先前工作中考虑的在两成分混合模型中估计未知零对称分布的问题。当使用位移位置和混合概率的一致估计量时，我们证明混合密度和未知零对称分量的非参数对数凹面最大似然估计值（MLE）在Hellinger距离中是一致的。如果移位位置和混合概率的估计量是$ \ sqrt n $一致，则我们确定这些MLE在$ L_1 $距离内以比率$ n ^ {-2/5} $收敛到真值。为了估计移动位置和混合概率，我们使用\ cite {hunteretal2007}提出的估计器。使用\ proglang {R}包\ pkg {logcondens.mode}可有效地计算未知的零对称密度。
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