## The Annals of Statistics

### Global rates of convergence of the MLEs of log-concave and $s$-concave densities

#### 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 $s<-1$.

#### Article information

Source
Ann. Statist., Volume 44, Number 3 (2016), 954-981.

Dates
Revised: September 2015
First available in Project Euclid: 11 April 2016

https://projecteuclid.org/euclid.aos/1460381683

Digital Object Identifier
doi:10.1214/15-AOS1394

Mathematical Reviews number (MathSciNet)
MR3485950

Zentralblatt MATH identifier
1338.62101

Subjects
Primary: 62G07: Density estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Doss, Charles R.; Wellner, Jon A. Global rates of convergence of the MLEs of log-concave and $s$-concave densities. Ann. Statist. 44 (2016), no. 3, 954--981. doi:10.1214/15-AOS1394. https://projecteuclid.org/euclid.aos/1460381683

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