Translator Disclaimer
June 2012 On statistical properties of sets fulfilling rolling-type conditions
A. Cuevas, R. Fraiman, B. Pateiro-López
Author Affiliations +
Adv. in Appl. Probab. 44(2): 311-329 (June 2012). DOI: 10.1239/aap/1339878713

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)).

Citation

Download Citation

A. Cuevas. R. Fraiman. B. Pateiro-López. "On statistical properties of sets fulfilling rolling-type conditions." Adv. in Appl. Probab. 44 (2) 311 - 329, June 2012. https://doi.org/10.1239/aap/1339878713

Information

Published: June 2012
First available in Project Euclid: 16 June 2012

zbMATH: 1252.47089
MathSciNet: MR2977397
Digital Object Identifier: 10.1239/aap/1339878713

Subjects:
Primary: 47N30
Secondary: 60D05, 62G05

Rights: Copyright © 2012 Applied Probability Trust

JOURNAL ARTICLE
19 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.44 • No. 2 • June 2012
Back to Top