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

Author

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  • Ery Arias-Castro
  • Beatriz Pateiro-López
  • Alberto Rodríguez-Casal

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.

Suggested Citation

  • Ery Arias-Castro & Beatriz Pateiro-López & Alberto Rodríguez-Casal, 2019. "Minimax Estimation of the Volume of a Set Under the Rolling Ball Condition," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1162-1173, July.
    • Handle: RePEc:taf:jnlasa:v:114:y:2019:i:527:p:1162-1173
    DOI: 10.1080/01621459.2018.1482751
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