  We study nonparametric maximum likelihood estimation of a log-concave
probability density and its distribution and hazard function. Some general
properties of these estimators are derived from two characterizations. 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
$(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference
between the empirical and estimated distribution function vanishes with rate
$o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.
