Shape-constrained density estimation with Wasserstein projection

TL;DR

This paper introduces a novel approach for univariate nonparametric density estimation using Wasserstein projection under shape constraints, providing structural insights and practical algorithms.

Contribution

It develops a convex optimization framework for Wasserstein projection estimation under shape constraints like monotonicity and log-concavity, with theoretical and computational advancements.

Findings

Structural properties of Wasserstein projection estimators are established.

A discretization method suitable for standard solvers is proposed.

Comparison with maximum likelihood estimators highlights differences and advantages.

Abstract

Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained density estimation via projection with respect to the $p$-Wasserstein distance, with a focus on the quadratic case $p = 2$. By considering shape constraints given by displacement convex subsets of the Wasserstein space, Wasserstein projection estimation is a convex optimization problem. We focus on two fundamental examples, namely non-increasing densities on $\mathbb{R}_+ := [0, \infty)$ and log-concave densities on $\mathbb{R}$. In each case, we prove structural properties of the Wasserstein projection estimator, propose a discretization which can be implemented by off-the-shelf solvers, and compare the projection estimator with the corresponding maximum likelihood estimator.