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Minimax Estimation of the Volume of a Set with Smooth Boundary (1605.01333v1)
Published 4 May 2016 in math.ST, stat.ME, and stat.TH
Abstract: We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume the domain has a boundary with positive reach. We propose a data splitting approach to correct the bias of the plug-in estimator based on the sample alpha-convex hull. We show that this simple estimator achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings.