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 R. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{2/5}$ when $-1/5 < s < \infty$ where $s=0$ corresponds to the log-concave case.