Methods for Estimation of Convex Sets

TL;DR

This paper reviews various methods for estimating convex sets in statistical shape constrained problems, highlighting differences in techniques and results under different metrics, and discussing computational challenges in high-dimensional settings.

Contribution

It provides a comprehensive overview of convex set estimation methods, comparing approaches under Nikodym and Hausdorff metrics, and addresses computational issues in high dimensions.

Findings

Different metrics require distinct estimation techniques.

Estimation under Hausdorff metric differs significantly from Nikodym.

Computational challenges increase with dimension.

Abstract

In the framework of shape constrained estimation, we review methods and works done in convex set estimation. These methods mostly build on stochastic and convex geometry, empirical process theory, functional analysis, linear programming, extreme value theory, etc. The statistical problems that we review include density support estimation, estimation of the level sets of densities or depth functions, nonparametric regression, etc. We focus on the estimation of convex sets under the Nikodym and Hausdorff metrics, which require different techniques and, quite surprisingly, lead to very different results, in particular in density support estimation. Finally, we discuss computational issues in high dimensions.