Abstract
We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1<s<\infty$ 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 $s<-1$.
Citation
Charles R. Doss. Jon A. Wellner. "Global rates of convergence of the MLEs of log-concave and $s$-concave densities." Ann. Statist. 44 (3) 954 - 981, June 2016. https://doi.org/10.1214/15-AOS1394
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