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(.,.). 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) + epsilon, where X and epsilon are independent, mu(.) belongs to a certain class of regression functions while E is a random error with log-concave density and mean zero.