The Annals of Statistics

Global rates of convergence in log-concave density estimation

Arlene K. H. Kim and Richard J. Samworth

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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.

Article information

Ann. Statist., Volume 44, Number 6 (2016), 2756-2779.

Received: April 2014
Revised: March 2016
First available in Project Euclid: 23 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties

Bracketing entropy density estimation global loss function log-concavity maximum likelihood estimation


Kim, Arlene K. H.; Samworth, Richard J. Global rates of convergence in log-concave density estimation. Ann. Statist. 44 (2016), no. 6, 2756--2779. doi:10.1214/16-AOS1480.

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Supplemental materials

  • Supplementary material to “Global rates of convergence in log-concave density estimation”. Proof of Theorem 1 and auxiliary results.