GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND s-CONCAVE DENSITIES

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on .. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when .1 < s < . where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s <.1.