Maximum likelihood estimation of a

We study nonparametric maximum likelihood estimation of a log-concave probability density and its dis tribution and hazard function. Some general properties of these estimators are derived from two charac terizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(?)/w)1//3 and typically (log(?)/rc)2/5, whereas the difference between the empirical and estimated distribution function vanishes with rate Op(n~1^) under certain regularity assumptions.