Statistical Science

Methods for Estimation of Convex Sets

Victor-Emmanuel Brunel

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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.

Article information

Source
Statist. Sci., Volume 33, Number 4 (2018), 615-632.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1543482061

Digital Object Identifier
doi:10.1214/18-STS669

Mathematical Reviews number (MathSciNet)
MR3881211

Zentralblatt MATH identifier
07032832

Keywords
Convex body set estimation Nikodym metric Hausdorff metric support function

Citation

Brunel, Victor-Emmanuel. Methods for Estimation of Convex Sets. Statist. Sci. 33 (2018), no. 4, 615--632. doi:10.1214/18-STS669. https://projecteuclid.org/euclid.ss/1543482061


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