On statistical properties of sets fulfilling rolling-type conditions

This is the peer reviewed version of the following article: Advances in Applied Probability 44.2 (2012): 311-329, which has been published in final form at http://dx.doi.org/10.1239/aap/1339878713

Detalles Bibliográficos
Autores: Cuevas González, Antonio, Fraiman, Ricardo, Pateiro-López, Beatriz
Tipo de recurso: artículo
Fecha de publicación:2012
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/669852
Acceso en línea:http://hdl.handle.net/10486/669852
https://dx.doi.org/10.1239/aap/1339878713
Access Level:acceso abierto
Palabra clave:R-convexity
Positive reach
Rolling condition
Glivenko-Cantelli classes
Set estimation
Boundary length
Excess mass
Matemáticas
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spelling On statistical properties of sets fulfilling rolling-type conditionsCuevas González, AntonioFraiman, RicardoPateiro-López, BeatrizR-convexityPositive reachRolling conditionGlivenko-Cantelli classesSet estimationBoundary lengthExcess massMatemáticasThis is the peer reviewed version of the following article: Advances in Applied Probability 44.2 (2012): 311-329, which has been published in final form at http://dx.doi.org/10.1239/aap/1339878713Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we obtain for the estimation of sets fulfilling a rolling condition 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 (a.s.) to that of the underlying support. Fifth, the above results are applied to get new consistency statements for level set estimators based on the excess mass methodology (Polonik, 1995)This work has been partially supported by Spanish Grants MTM2010-17366 and CCG10-UAM/ESP-5494 (A. Cuevas), MTM2010-17366 and Argentinian grant PIC-2008-0921 (R. Fraiman) and by Spanish Grant MTM2008-03010 and the IAP research network grant no. P6/03 from the Belgian government (B. Pateiro-López)Applied Probability TrustDepartamento de MatemáticasFacultad de Ciencias20122012-06-01research articlehttp://purl.org/coar/resource_type/c_2df8fbb1AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/669852https://dx.doi.org/10.1239/aap/1339878713reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/6698522026-06-23T12:46:27Z
dc.title.none.fl_str_mv On statistical properties of sets fulfilling rolling-type conditions
title On statistical properties of sets fulfilling rolling-type conditions
spellingShingle On statistical properties of sets fulfilling rolling-type conditions
Cuevas González, Antonio
R-convexity
Positive reach
Rolling condition
Glivenko-Cantelli classes
Set estimation
Boundary length
Excess mass
Matemáticas
title_short On statistical properties of sets fulfilling rolling-type conditions
title_full On statistical properties of sets fulfilling rolling-type conditions
title_fullStr On statistical properties of sets fulfilling rolling-type conditions
title_full_unstemmed On statistical properties of sets fulfilling rolling-type conditions
title_sort On statistical properties of sets fulfilling rolling-type conditions
dc.creator.none.fl_str_mv Cuevas González, Antonio
Fraiman, Ricardo
Pateiro-López, Beatriz
author Cuevas González, Antonio
author_facet Cuevas González, Antonio
Fraiman, Ricardo
Pateiro-López, Beatriz
author_role author
author2 Fraiman, Ricardo
Pateiro-López, Beatriz
author2_role author
author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
dc.subject.none.fl_str_mv R-convexity
Positive reach
Rolling condition
Glivenko-Cantelli classes
Set estimation
Boundary length
Excess mass
Matemáticas
topic R-convexity
Positive reach
Rolling condition
Glivenko-Cantelli classes
Set estimation
Boundary length
Excess mass
Matemáticas
description This is the peer reviewed version of the following article: Advances in Applied Probability 44.2 (2012): 311-329, which has been published in final form at http://dx.doi.org/10.1239/aap/1339878713
publishDate 2012
dc.date.none.fl_str_mv 2012
2012-06-01
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/669852
https://dx.doi.org/10.1239/aap/1339878713
url http://hdl.handle.net/10486/669852
https://dx.doi.org/10.1239/aap/1339878713
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Applied Probability Trust
publisher.none.fl_str_mv Applied Probability Trust
dc.source.none.fl_str_mv reponame:Biblos-e Archivo. Repositorio Institucional de la UAM
instname:Universidad Autónoma de Madrid
instname_str Universidad Autónoma de Madrid
reponame_str Biblos-e Archivo. Repositorio Institucional de la UAM
collection Biblos-e Archivo. Repositorio Institucional de la UAM
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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