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June 2016 Global rates of convergence of the MLEs of log-concave and $s$-concave densities
Charles R. Doss, Jon A. Wellner
Ann. Statist. 44(3): 954-981 (June 2016). DOI: 10.1214/15-AOS1394


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 $s<-1$.


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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.


Received: 1 June 2013; Revised: 1 September 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1338.62101
MathSciNet: MR3485950
Digital Object Identifier: 10.1214/15-AOS1394

Primary: 62G07
Secondary: 62G05 , 62G20

Keywords: $s$-concave , Bracketing entropy , consistency , Empirical processes , global rate , Hellinger metric , log-concave

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • June 2016
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