The Annals of Statistics

Adaptive estimation of planar convex sets

T. Tony Cai, Adityanand Guntuboyina, and Yuting Wei

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In this paper, we consider adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function. Both the problem of estimating the support function at a point and that of estimating the whole convex set are studied. For pointwise estimation, we consider the problem in a general nonasymptotic framework, which evaluates the performance of a procedure at each individual set, instead of the worst case performance over a large parameter space as in conventional minimax theory. A data-driven adaptive estimator is proposed and is shown to be optimally adaptive to every compact, convex set. For estimating the whole convex set, we propose estimators that are shown to adaptively achieve the optimal rate of convergence. In both of these problems, our analysis makes no smoothness assumptions on the boundary of the unknown convex set.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1018-1049.

Received: June 2016
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

Adaptive estimation circle convexity convex set Hausdorff distance minimax rate of convergence support function


Cai, T. Tony; Guntuboyina, Adityanand; Wei, Yuting. Adaptive estimation of planar convex sets. Ann. Statist. 46 (2018), no. 3, 1018--1049. doi:10.1214/17-AOS1576.

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Supplemental materials

  • Supplement to “Adaptive estimation of planar convex sets”. Technical Appendix. Contains proofs of some results in the main paper as well as additional technical results and simulations.