On Statistical Properties of Sets Fulfilling Rolling-Type Conditions

Author:

Cuevas A.,Fraiman R.,Pateiro-López B.

Abstract

Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of ‘full consistency’ (i.e. consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics,Statistics and Probability

Cited by 10 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Computable Bounds for the Reach and r-Convexity of Subsets of $${{\mathbb {R}}}^d$$;Discrete & Computational Geometry;2024-01-27

2. Home-range estimation under a restricted sample scheme;Journal of Nonparametric Statistics;2023-11-10

3. Statistical analysis of measures of non-convexity;TEST;2023-10-12

4. Universally consistent estimation of the reach;Journal of Statistical Planning and Inference;2023-07

5. A data-adaptive method for estimating density level sets under shape conditions;The Annals of Statistics;2022-06-01

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