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June 2018 Adaptive estimation of planar convex sets
T. Tony Cai, Adityanand Guntuboyina, Yuting Wei
Ann. Statist. 46(3): 1018-1049 (June 2018). DOI: 10.1214/17-AOS1576


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.


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T. Tony Cai. Adityanand Guntuboyina. Yuting Wei. "Adaptive estimation of planar convex sets." Ann. Statist. 46 (3) 1018 - 1049, June 2018.


Received: 1 June 2016; Published: June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 06897921
MathSciNet: MR3797995
Digital Object Identifier: 10.1214/17-AOS1576

Primary: 62G07
Secondary: 52A20

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • June 2018
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