Universally consistent estimation of the reach

Cholaquidis, Alejandro - Fraiman, Ricardo - Moreno, Leonardo

Resumen:

The reach of a set M⊂Rd, also known as condition number when M is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when M is a manifold.


Detalles Bibliográficos
2022
ANII: FCE_1_2019_1_156054
Mathematics - Statistics theory
Inglés
Universidad de la República
COLIBRI
https://hdl.handle.net/20.500.12008/37376
Acceso abierto
Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
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author Cholaquidis, Alejandro
author2 Fraiman, Ricardo
Moreno, Leonardo
author2_role author
author
author_facet Cholaquidis, Alejandro
Fraiman, Ricardo
Moreno, Leonardo
author_role author
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dc.contributor.filiacion.none.fl_str_mv Cholaquidis Alejandro, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.
Fraiman Ricardo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.
Moreno Leonardo, Universidad de la República (Uruguay). FCEA
dc.creator.none.fl_str_mv Cholaquidis, Alejandro
Fraiman, Ricardo
Moreno, Leonardo
dc.date.accessioned.none.fl_str_mv 2023-06-02T14:29:19Z
dc.date.available.none.fl_str_mv 2023-06-02T14:29:19Z
dc.date.issued.none.fl_str_mv 2022
dc.description.abstract.none.fl_txt_mv The reach of a set M⊂Rd, also known as condition number when M is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when M is a manifold.
dc.description.es.fl_txt_mv Publicado también en: Journal of Statistical Planning and Inference, 2023, 225: 110-120. DOI: 10.1016/j.jspi.2022.11.007
dc.description.sponsorship.none.fl_txt_mv ANII: FCE_1_2019_1_156054
dc.format.extent.es.fl_str_mv 15 h
dc.format.mimetype.es.fl_str_mv application/pdf
dc.identifier.citation.es.fl_str_mv Cholaquidis, A, Fraiman, R y Moreno, L. "Universally consistent estimation of the reach". [Preprint]. Publicado en: Mathematics (Statistics Theory). 2022, arXiv:2110.12208, Nov 2022. 15 h.
dc.identifier.doi.none.fl_str_mv 10.48550/arXiv.2110.12208
dc.identifier.uri.none.fl_str_mv https://hdl.handle.net/20.500.12008/37376
dc.language.iso.none.fl_str_mv en
eng
dc.relation.ispartof.es.fl_str_mv Mathematics (Statistics Theory), arXiv:2110.12208, Nov 2022.
dc.rights.license.none.fl_str_mv Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
dc.source.none.fl_str_mv reponame:COLIBRI
instname:Universidad de la República
instacron:Universidad de la República
dc.subject.es.fl_str_mv Mathematics - Statistics theory
dc.title.none.fl_str_mv Universally consistent estimation of the reach
dc.type.es.fl_str_mv Preprint
dc.type.none.fl_str_mv info:eu-repo/semantics/preprint
dc.type.version.none.fl_str_mv info:eu-repo/semantics/submittedVersion
description Publicado también en: Journal of Statistical Planning and Inference, 2023, 225: 110-120. DOI: 10.1016/j.jspi.2022.11.007
eu_rights_str_mv openAccess
format preprint
id COLIBRI_5e4bc48ff5945e6981aef19c37667e6e
identifier_str_mv Cholaquidis, A, Fraiman, R y Moreno, L. "Universally consistent estimation of the reach". [Preprint]. Publicado en: Mathematics (Statistics Theory). 2022, arXiv:2110.12208, Nov 2022. 15 h.
10.48550/arXiv.2110.12208
instacron_str Universidad de la República
institution Universidad de la República
instname_str Universidad de la República
language eng
language_invalid_str_mv en
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publishDate 2022
reponame_str COLIBRI
repository.mail.fl_str_mv mabel.seroubian@seciu.edu.uy
repository.name.fl_str_mv COLIBRI - Universidad de la República
repository_id_str 4771
rights_invalid_str_mv Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)
spelling Cholaquidis Alejandro, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.Fraiman Ricardo, Universidad de la República (Uruguay). Facultad de Ciencias. Centro de Matemática.Moreno Leonardo, Universidad de la República (Uruguay). FCEA2023-06-02T14:29:19Z2023-06-02T14:29:19Z2022Cholaquidis, A, Fraiman, R y Moreno, L. "Universally consistent estimation of the reach". [Preprint]. Publicado en: Mathematics (Statistics Theory). 2022, arXiv:2110.12208, Nov 2022. 15 h.https://hdl.handle.net/20.500.12008/3737610.48550/arXiv.2110.12208Publicado también en: Journal of Statistical Planning and Inference, 2023, 225: 110-120. DOI: 10.1016/j.jspi.2022.11.007The reach of a set M⊂Rd, also known as condition number when M is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when M is a manifold.Submitted by Faget Cecilia (lfaget@fcien.edu.uy) on 2023-06-02T12:06:09Z No. of bitstreams: 2 license_rdf: 23149 bytes, checksum: 1996b8461bc290aef6a27d78c67b6b52 (MD5) 2110.12208.pdf: 569372 bytes, checksum: 6c2d497e6e43e9080c44ffb689a9bf4c (MD5)Approved for entry into archive by Faget Cecilia (lfaget@fcien.edu.uy) on 2023-06-02T13:51:16Z (GMT) No. of bitstreams: 2 license_rdf: 23149 bytes, checksum: 1996b8461bc290aef6a27d78c67b6b52 (MD5) 2110.12208.pdf: 569372 bytes, checksum: 6c2d497e6e43e9080c44ffb689a9bf4c (MD5)Made available in DSpace by Luna Fabiana (fabiana.luna@seciu.edu.uy) on 2023-06-02T14:29:19Z (GMT). No. of bitstreams: 2 license_rdf: 23149 bytes, checksum: 1996b8461bc290aef6a27d78c67b6b52 (MD5) 2110.12208.pdf: 569372 bytes, checksum: 6c2d497e6e43e9080c44ffb689a9bf4c (MD5) Previous issue date: 2022ANII: FCE_1_2019_1_15605415 happlication/pdfenengMathematics (Statistics Theory), arXiv:2110.12208, Nov 2022.Las obras depositadas en el Repositorio se rigen por la Ordenanza de los Derechos de la Propiedad Intelectual de la Universidad de la República.(Res. Nº 91 de C.D.C. de 8/III/1994 – D.O. 7/IV/1994) y por la Ordenanza del Repositorio Abierto de la Universidad de la República (Res. Nº 16 de C.D.C. de 07/10/2014)info:eu-repo/semantics/openAccessLicencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0)Mathematics - Statistics theoryUniversally consistent estimation of the reachPreprintinfo:eu-repo/semantics/preprintinfo:eu-repo/semantics/submittedVersionreponame:COLIBRIinstname:Universidad de la Repúblicainstacron:Universidad de la RepúblicaCholaquidis, AlejandroFraiman, RicardoMoreno, LeonardoLICENSElicense.txtlicense.txttext/plain; charset=utf-84267http://localhost:8080/xmlui/bitstream/20.500.12008/37376/5/license.txt6429389a7df7277b72b7924fdc7d47a9MD55CC-LICENSElicense_urllicense_urltext/plain; charset=utf-850http://localhost:8080/xmlui/bitstream/20.500.12008/37376/2/license_urla006180e3f5b2ad0b88185d14284c0e0MD52license_textlicense_texttext/html; charset=utf-838782http://localhost:8080/xmlui/bitstream/20.500.12008/37376/3/license_texte8c30e04e865334cac2bfcba70aad8cbMD53license_rdflicense_rdfapplication/rdf+xml; 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- Universidad de la Repúblicafalse
spellingShingle Universally consistent estimation of the reach
Cholaquidis, Alejandro
Mathematics - Statistics theory
status_str submittedVersion
title Universally consistent estimation of the reach
title_full Universally consistent estimation of the reach
title_fullStr Universally consistent estimation of the reach
title_full_unstemmed Universally consistent estimation of the reach
title_short Universally consistent estimation of the reach
title_sort Universally consistent estimation of the reach
topic Mathematics - Statistics theory
url https://hdl.handle.net/20.500.12008/37376

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