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Unbiased estimation of the volume of a convex body

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  • Baldin, Nikolay
  • Reiß, Markus

Abstract

Based on observations of points uniformly distributed over a convex set in Rd, 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 adjusted set estimator for the convex body itself is proposed.

Suggested Citation

  • Baldin, Nikolay & Reiß, Markus, 2016. "Unbiased estimation of the volume of a convex body," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3716-3732.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:12:p:3716-3732
    DOI: 10.1016/j.spa.2016.04.014
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    References listed on IDEAS

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    1. Baddeley, Adrian & Turner, Rolf, 2005. "spatstat: An R Package for Analyzing Spatial Point Patterns," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 12(i06).
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    Cited by:

    1. Jens Grygierek, 2021. "Multivariate Normal Approximation for Functionals of Random Polytopes," Journal of Theoretical Probability, Springer, vol. 34(2), pages 897-922, June.

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