Statistical Science

Shape Constrained Density Estimation Via Penalized Rényi Divergence

Roger Koenker and Ivan Mizera

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Abstract

Shape constraints play an increasingly prominent role in nonparametric function estimation. While considerable recent attention has been focused on log concavity as a regularizing device in nonparametric density estimation, weaker forms of concavity constraints encompassing larger classes of densities have received less attention but offer some additional flexibility. Heavier tail behavior and sharper modal peaks are better adapted to such weaker concavity constraints. When paired with appropriate maximal entropy estimation criteria, these weaker constraints yield tractable, convex optimization problems that broaden the scope of shape constrained density estimation in a variety of applied subject areas.

In contrast to our prior work, Koenker and Mizera [Ann. Statist. 38 (2010) 2998–3027], that focused on the log concave ($\alpha=1$) and Hellinger ($\alpha=1/2$) constraints, here we describe methods enabling imposition of even weaker, $\alpha\leq0$ constraints. An alternative formulation of the concavity constraints for densities in dimension $d\geq2$ also significantly expands the applicability of our proposed methods for multivariate data. Finally, we illustrate the use of the Rényi divergence criterion for norm-constrained estimation of densities in the absence of a shape constraint.

Article information

Source
Statist. Sci., Volume 33, Number 4 (2018), 510-526.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1543482056

Digital Object Identifier
doi:10.1214/18-STS658

Mathematical Reviews number (MathSciNet)
MR3881206

Zentralblatt MATH identifier
07032827

Keywords
Density estimation shape constraints Rényi entropy convex optimization

Citation

Koenker, Roger; Mizera, Ivan. Shape Constrained Density Estimation Via Penalized Rényi Divergence. Statist. Sci. 33 (2018), no. 4, 510--526. doi:10.1214/18-STS658. https://projecteuclid.org/euclid.ss/1543482056


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