Approximation by log-concave distributions, with applications to regression

We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback–Leibler-type functional. We show that such an approximation exists if and only if $P$ has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on $P$ with respect to Mallows distance $D_1(\cdot ,\cdot )$. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response $Y=\mu (X)+ϵ$, where $X$ and $ϵ$ are independent, $\mu (\cdot )$ belongs to a certain class of regression functions while $ϵ$ is a random error with log-concave density and mean zero.