Data-based decision rules about the convexity of the support of a distribution

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• dc.contributor.author Delicado, Pedro
• dc.contributor.author Hernández, Adolfo
• dc.contributor.author Lugosi, Gábor
• dc.date.accessioned 2023-12-12T06:41:03Z
• dc.date.available 2023-12-12T06:41:03Z
• dc.date.issued 2014
• dc.description.abstract Given n independent, identically distributed random vectors in Rd, drawn from a common density f, one wishes to find out whether the support of f is convex or not. In this paper we describe a decision rule which decides correctly for sufficiently large n, with probability 1, whenever f is bounded away from zero in its compact support. We also show that the assumption of boundedness is necessary. The rule is based on a statistic that is a second-order U-statistic with a random kernel. Moreover, we suggest a way of approximating the distribution of the statistic under the hypothesis of convexity of the support. The performance of the proposed method is illustrated on simulated data sets. As an example of its potential statistical implications, the decision rule is used to automatically choose the tuning parameter of ISOMAP, a nonlinear dimensionality reduction method.
• dc.description.sponsorship Supported by Spanish Ministry of Education and Science and FEDER (MTM2010-14887). Supported by the Spanish Ministry of Science and Technology grant MTM2012-37195.
• dc.format.mimetype application/pdf
• dc.identifier.citation Delicado P, Hernández A, Lugosi G. Data-based decision rules about the convexity of the support of a distribution. Electron J Statist. 2014;8(1):96-129. DOI: 10.1214/14-EJS877
• dc.identifier.doi http://dx.doi.org/10.1214/14-EJS877
• dc.identifier.issn 1935-7524
• dc.identifier.uri http://hdl.handle.net/10230/58507
• dc.language.iso eng
• dc.publisher Institute of Mathematical Statistics
• dc.relation.ispartof Electronic Journal of Statistics. 2014;8(1):96-129
• dc.relation.projectID info:eu-repo/grantAgreement/ES/3PN/MTM2010-14887
• dc.relation.projectID info:eu-repo/grantAgreement/ES/3PN/MTM2012-37195
• dc.rights Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society. Copyright for all articles in EJP is CC BY 4.0.
• dc.rights.accessRights info:eu-repo/semantics/openAccess
• dc.rights.uri http://creativecommons.org/licenses/by/4.0/deed.es
• dc.subject.keyword Bootstrap subsampling
• dc.subject.keyword Dimensionality reduction
• dc.subject.keyword Discernibility between hypotheses
• dc.subject.keyword ISOMAP
• dc.subject.keyword Set estimation
• dc.subject.keyword U-statistics
• dc.title Data-based decision rules about the convexity of the support of a distribution
• dc.type info:eu-repo/semantics/article
• dc.type.version info:eu-repo/semantics/publishedVersion