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 $Y=μ(X)+ε$, where $X$ and $ε$ are independent, $μ(\cdot)$ belongs to a certain class of regression functions while $ε$ is a random error with log-concave density and mean zero.

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