Unbiased estimation of the volume of a convex body

2015/02/19 by Nikolay Baldin, Baldin, Nikolay, Markus Reiß +1 · 1 citation
Mathematics · #60G55 #62G05 #62M30 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:60G55 #msc:62G05 #msc:62M30 #stat.TH

paper · pdf · doi:10.48550/arxiv.1502.05510

slightly extended version

arxiv created 2016/01/21 · arxiv updated 2016/01/22

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.

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