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December 2016 Global rates of convergence in log-concave density estimation
Arlene K. H. Kim, Richard J. Samworth
Ann. Statist. 44(6): 2756-2779 (December 2016). DOI: 10.1214/16-AOS1480

Abstract

The estimation of a log-concave density on $\mathbb{R}^{d}$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order $n^{-4/5}$, when $d=1$, and order $n^{-2/(d+1)}$ when $d\geq2$. In particular, this reveals a sense in which, when $d\geq3$, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared). Second, we show that for $d\leq3$, the Hellinger $\varepsilon$-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like $\max\{\varepsilon^{-d/2},\varepsilon^{-(d-1)}\}$ (up to a logarithmic factor when $d=2$). This enables us to prove that when $d\leq3$ the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when $d=2,3$) with respect to squared Hellinger loss.

Citation

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Arlene K. H. Kim. Richard J. Samworth. "Global rates of convergence in log-concave density estimation." Ann. Statist. 44 (6) 2756 - 2779, December 2016. https://doi.org/10.1214/16-AOS1480

Information

Received: 1 April 2014; Revised: 1 March 2016; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1360.62157
MathSciNet: MR3576560
Digital Object Identifier: 10.1214/16-AOS1480

Subjects:
Primary: 62G07 , 62G20

Keywords: Bracketing entropy , Density estimation , global loss function , Log-concavity , maximum likelihood estimation

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • December 2016
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