ON POINCARÉ CONE PROPERTY

A domain S⊂Rd is said to fulfill the Poincaré cone property if any point in the boundary of S is the vertex of a (finite) cone which does not otherwise intersects the closure S¯. For more than a century, this condition has played a relevant role in the theory of partial differential equations, as a...

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Detalles Bibliográficos
Autores: Cholaquidis, Alejandro, Cuevas González, Antonio, Fraiman, Ricardo
Tipo de recurso: artículo
Fecha de publicación:2014
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/669371
Acceso en línea:http://hdl.handle.net/10486/669371
https://dx.doi.org/10.1214/13-AOS1188
Access Level:acceso abierto
Palabra clave:Poincaré property
Glivenko–Cantelli classes
Set estimation
Matemáticas
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spelling ON POINCARÉ CONE PROPERTYCholaquidis, AlejandroCuevas González, AntonioFraiman, RicardoPoincaré propertyGlivenko–Cantelli classesSet estimationMatemáticasA domain S⊂Rd is said to fulfill the Poincaré cone property if any point in the boundary of S is the vertex of a (finite) cone which does not otherwise intersects the closure S¯. For more than a century, this condition has played a relevant role in the theory of partial differential equations, as a shape assumption aimed to ensure the existence of a solution for the classical Dirichlet problem on S. In a completely different setting, this paper is devoted to analyze some statistical applications of the Poincaré cone property (when defined in a slightly stronger version). First, we show that this condition can be seen as a sort of generalized convexity: while it is considerably less restrictive than convexity, it still retains some “convex flavor”. In particular, when imposed to a probability support S, this property allows the estimation of S from a random sample of points, using the “hull principle” much in the same way as a convex support is estimated using the convex hull of the sample points. The statistical properties of such hull estimator (consistency, convergence rates, boundary estimation) are considered in detail. Second, it is shown that the class of sets fulfilling the Poincaré property is a P-Glivenko-Cantelli class for any absolutely continuous distribution P on Rd. This has some independent interest in the theory of empirical processes, since it extends the classical analogous result, established for convex sets, to a much larger class. Third, an algorithm to approximate the cone-convex hull of a finite sample of points is proposed and some practical illustrations are givenSupported in part by Spanish Grant MTM2010-17366Institute of Mathematical StatisticsDepartamento de MatemáticasFacultad de Ciencias20142014-03-21research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/669371https://dx.doi.org/10.1214/13-AOS1188reponame: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:10486/6693712026-06-23T12:46:27Z
dc.title.none.fl_str_mv ON POINCARÉ CONE PROPERTY
title ON POINCARÉ CONE PROPERTY
spellingShingle ON POINCARÉ CONE PROPERTY
Cholaquidis, Alejandro
Poincaré property
Glivenko–Cantelli classes
Set estimation
Matemáticas
title_short ON POINCARÉ CONE PROPERTY
title_full ON POINCARÉ CONE PROPERTY
title_fullStr ON POINCARÉ CONE PROPERTY
title_full_unstemmed ON POINCARÉ CONE PROPERTY
title_sort ON POINCARÉ CONE PROPERTY
dc.creator.none.fl_str_mv Cholaquidis, Alejandro
Cuevas González, Antonio
Fraiman, Ricardo
author Cholaquidis, Alejandro
author_facet Cholaquidis, Alejandro
Cuevas González, Antonio
Fraiman, Ricardo
author_role author
author2 Cuevas González, Antonio
Fraiman, Ricardo
author2_role author
author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
dc.subject.none.fl_str_mv Poincaré property
Glivenko–Cantelli classes
Set estimation
Matemáticas
topic Poincaré property
Glivenko–Cantelli classes
Set estimation
Matemáticas
description A domain S⊂Rd is said to fulfill the Poincaré cone property if any point in the boundary of S is the vertex of a (finite) cone which does not otherwise intersects the closure S¯. For more than a century, this condition has played a relevant role in the theory of partial differential equations, as a shape assumption aimed to ensure the existence of a solution for the classical Dirichlet problem on S. In a completely different setting, this paper is devoted to analyze some statistical applications of the Poincaré cone property (when defined in a slightly stronger version). First, we show that this condition can be seen as a sort of generalized convexity: while it is considerably less restrictive than convexity, it still retains some “convex flavor”. In particular, when imposed to a probability support S, this property allows the estimation of S from a random sample of points, using the “hull principle” much in the same way as a convex support is estimated using the convex hull of the sample points. The statistical properties of such hull estimator (consistency, convergence rates, boundary estimation) are considered in detail. Second, it is shown that the class of sets fulfilling the Poincaré property is a P-Glivenko-Cantelli class for any absolutely continuous distribution P on Rd. This has some independent interest in the theory of empirical processes, since it extends the classical analogous result, established for convex sets, to a much larger class. Third, an algorithm to approximate the cone-convex hull of a finite sample of points is proposed and some practical illustrations are given
publishDate 2014
dc.date.none.fl_str_mv 2014
2014-03-21
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/669371
https://dx.doi.org/10.1214/13-AOS1188
url http://hdl.handle.net/10486/669371
https://dx.doi.org/10.1214/13-AOS1188
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 Institute of Mathematical Statistics
publisher.none.fl_str_mv Institute of Mathematical Statistics
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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