The Annals of Statistics

Approximation and estimation of s-concave densities via Rényi divergences

Qiyang Han and Jon A. Wellner

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In this paper, we study the approximation and estimation of $s$-concave densities via Rényi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave density exists and is unique via the procedure of minimizing a divergence functional proposed by [Ann. Statist. 38 (2010) 2998–3027] if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_{n}\to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_{1}$ metrics and uniformly on closed subsets of the continuity set of the limit. Moreover, directional derivatives of the projected densities also enjoy local uniform convergence. This contains both on-the-model and off-the-model situations, and entails strong consistency of the divergence estimator of an $s$-concave density under mild conditions. One interesting and important feature for the Rényi divergence estimator of an $s$-concave density is that the estimator is intrinsically related with the estimation of log-concave densities via maximum likelihood methods. In fact, we show that for $d=1$ at least, the Rényi divergence estimators for $s$-concave densities converge to the maximum likelihood estimator of a log-concave density as $s\nearrow0$. The Rényi divergence estimator shares similar characterizations as the MLE for log-concave distributions, which allows us to develop pointwise asymptotic distribution theory assuming that the underlying density is $s$-concave.

Article information

Ann. Statist., Volume 44, Number 3 (2016), 1332-1359.

Received: June 2015
Revised: October 2015
First available in Project Euclid: 11 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

$s$-concavity consistency projection asymptotic distribution mode estimation nonparametric estimation shape constraints


Han, Qiyang; Wellner, Jon A. Approximation and estimation of s -concave densities via Rényi divergences. Ann. Statist. 44 (2016), no. 3, 1332--1359. doi:10.1214/15-AOS1408.

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Supplemental materials

  • Supplement to “Approximation and estimation of $s$-concave densities via Rényi divergences”. In the supplement Han and Wellner (2015), we provide details of the omitted proofs for Sections 2, 3, 4 and 6 and some auxiliary results from convex analysis used in the main paper.