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June 2016 Approximation and estimation of s-concave densities via Rényi divergences
Qiyang Han, Jon A. Wellner
Ann. Statist. 44(3): 1332-1359 (June 2016). DOI: 10.1214/15-AOS1408

Abstract

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.

Citation

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Qiyang Han. Jon A. Wellner. "Approximation and estimation of s-concave densities via Rényi divergences." Ann. Statist. 44 (3) 1332 - 1359, June 2016. https://doi.org/10.1214/15-AOS1408

Information

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

zbMATH: 1338.62105
MathSciNet: MR3485962
Digital Object Identifier: 10.1214/15-AOS1408

Subjects:
Primary: 62G07 , 62H12
Secondary: 62G05 , 62G20

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • June 2016
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