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 < 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.
Bibliographical notePublisher Copyright:
© Institute of Mathematical Statistics, 2016.
- Bracketing entropy
- Empirical processes
- Global rate
- Hellinger metric