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


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)).


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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.


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

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

Primary: 47N30
Secondary: 60D05 , 62G05

Keywords: boundary length , excess mass , Glivenko-Cantelli class , positive reach , r-convexity , rolling condition , set estimation

Rights: Copyright © 2012 Applied Probability Trust


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Vol.44 • No. 2 • June 2012
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