Approximation by Log - Concave Distributions with Applications to Regression

We study the approximation of arbitrary distributions P on ddimensional 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 D1(·, ·). 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) + , where X and are independent, μ(·) belongs to a certain class of regression functions while is a random error with log-concave density.