Unbiased estimation of the volume of a convex body

TL;DR

This paper introduces a new, nearly unbiased, minimax optimal estimator for the volume of a convex body in , based on uniform point observations, outperforming previous methods and including an improved set estimator.

Contribution

It proposes a novel volume estimator using a Poisson point process model that is nearly unbiased, minimax optimal, and does not assume boundary conditions.

Findings

Estimator outperforms previous volume estimators in numerical studies.

The approach is minimax optimal and nearly unbiased with minimal variance.

An improved convex set estimator is also proposed.

Abstract

Based on observations of points uniformly distributed over a convex set in $\R^d$, a new estimator for the volume of the convex set is proposed. The estimator is minimax optimal and also efficient non-asymptotically: it is nearly unbiased with minimal variance among all unbiased oracle-type estimators. Our approach is based on a Poisson point process model and as an ingredient, we prove that the convex hull is a sufficient and complete statistic. No hypotheses on the boundary of the convex set are imposed. In a numerical study, we show that the estimator outperforms earlier estimators for the volume. In addition, an improved set estimator for the convex body itself is proposed.