Minimax Estimation of the Volume of a Set Under the Rolling Ball Condition

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 that 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 α-convex hull. We show that this simple estimator achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings. Supplementary materials for this article are available online.

KEYWORDS:

• Minimax lower bound
• r-Convex hull
• Rolling condition
• Support estimation
• Volume estimation

Additional information

Funding

This work was partially supported by the U.S. National Science Foundation (DMS 1513465) and by the Spanish Ministry of Economy and Competitiveness and ERDF funds (MTM2016-76969P).