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October 2009 Adaptive Hausdorff estimation of density level sets
Aarti Singh, Clayton Scott, Robert Nowak
Ann. Statist. 37(5B): 2760-2782 (October 2009). DOI: 10.1214/08-AOS661

Abstract

Consider the problem of estimating the γ-level set Gγ*={x: f(x)≥γ} of an unknown d-dimensional density function f based on n independent observations X1, …, Xn from the density. This problem has been addressed under global error criteria related to the symmetric set difference. However, in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere. The Hausdorff error criterion provides this degree of uniformity and, hence, is more appropriate in such situations. It is known that the minimax optimal rate of error convergence for the Hausdorff metric is (n/log n)−1/(d+2α) for level sets with boundaries that have a Lipschitz functional form, where the parameter α characterizes the regularity of the density around the level of interest. However, the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter α. Furthermore, previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets (e.g., the boundary fragment and star-shaped sets) that effectively reduce the set estimation problem to a function estimation problem. This characterization precludes level sets with multiple connected components, which are fundamental to many applications. This paper presents a fully data-driven procedure that is adaptive to unknown regularity conditions and achieves near minimax optimal Hausdorff error control for a class of density level sets with very general shapes and multiple connected components.

Citation

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Aarti Singh. Clayton Scott. Robert Nowak. "Adaptive Hausdorff estimation of density level sets." Ann. Statist. 37 (5B) 2760 - 2782, October 2009. https://doi.org/10.1214/08-AOS661

Information

Published: October 2009
First available in Project Euclid: 17 July 2009

zbMATH: 1173.62019
MathSciNet: MR2541446
Digital Object Identifier: 10.1214/08-AOS661

Subjects:
Primary: 62G05 , 62G20

Keywords: Adaptivity , Density level set , Hausdorff error , rates of convergence

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 5B • October 2009
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