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On Statistical Properties of Sets Fulfilling Rolling-Type Conditions

Part of: Nonparametric inference Geometric probability and stochastic geometry Miscellaneous applications of operator theory

Published online by Cambridge University Press:  04 January 2016

A. Cuevas ,
R. Fraiman  and
B. Pateiro-López
A. Cuevas*
Affiliation:
Universidad Autónoma de Madrid
R. Fraiman*
Affiliation:
Universidad de San Andrés and Universidad de la República
B. Pateiro-López*
Affiliation:
Universidad de Santiago de Compostela
*
∗ Postal address: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain.
∗∗ Postal address: Departamento de Matemática y Ciencias, Universidad de San Andrés, Buenos Aires, Argentina.
∗∗∗ Postal address: Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain. Email address: beatriz.pateiro@usc.es
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Abstract

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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)).

Keywords

r-convexitypositive reachrolling conditionGlivenko-Cantelli classset estimationboundary lengthexcess mass

MSC classification

Primary: 47N30: Applications in probability theory and statistics
Secondary: 60D05: Geometric probability and stochastic geometry 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 311 - 329
Copyright
© Applied Probability Trust 

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