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hal.structure.identifierLaboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
dc.contributor.authorDoumic, Marie
HAL ID: 2247
hal.structure.identifierLaboratoire d'Analyse et de Mathématiques Appliquées [LAMA]
hal.structure.identifierCentre de Recherche en Économie et Statistique [CREST]
dc.contributor.authorHoffmann, Marc*
hal.structure.identifierLaboratoire Jean Alexandre Dieudonné [JAD]
dc.contributor.authorReynaud-Bouret, Patricia
HAL ID: 8239
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorRivoirard, Vincent*
dc.subjectAggregation-fragmentation h equationsen
dc.subjectOracle inequalitiesen
dc.subjectCell-division equationen
dc.subjectNonparametric density estimationen
dc.subjectStatistical inverse problemsen
dc.subjectLepski methoden
dc.titleNonparametric estimation of the division rate of a size-structured populationen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherLaboratoire Jacques-Louis Lions (LJLL) CNRS : UMR7598 – Université Pierre et Marie Curie - Paris VI;France
dc.contributor.editoruniversityotherLaboratoire Jean Alexandre Dieudonné (JAD) CNRS : UMR6621 – Université de Nice Sophia-Antipolis;France
dc.contributor.editoruniversityotherCentre de Recherche en Économie et Statistique (CREST) INSEE – École Nationale de la Statistique et de l'Administration Économique;France
dc.contributor.editoruniversityotherBANG (INRIA Rocquencourt) INRIA – Ecole Normale Supérieure de Paris - ENS Paris;France
dc.contributor.editoruniversityotherLaboratoire d'Analyse et de Mathématiques Appliquées (LAMA) CNRS : UMR8050 – Université Paris-Est – Université Paris XII Val de Marne;France
dc.description.abstractenWe consider the problem of estimating the division rate of a size-structured population in a nonparametric setting. The size of the system evolves according to a transport-fragmentation equation: each individual grows with a given transport rate, and splits into two offsprings of the same size, following a binary fragmentation process with unknown division rate that depends on its size. In contrast to a deterministic inverse problem approach, as in (Perthame, Zubelli, 2007) and (Doumic, Perthame, Zubelli, 2009), we take in this paper the perspective of statistical inference: our data consists in a large sample of the size of individuals, when the evolution of the system is close to its time-asymptotic behavior, so that it can be related to the eigenproblem of the considered transport-fragmentation equation (see \cite{PR} for instance). By estimating statistically each term of the eigenvalue problem and by suitably inverting a certain linear operator (see previously quoted articles), we are able to construct a more realistic estimator of the division rate that achieves the same optimal error bound as in related deterministic inverse problems. Our procedure relies on kernel methods with automatic bandwidth selection. It is inspired by model selection and recent results of Goldenschluger and Lepski.en
dc.relation.isversionofjnlnameSIAM Journal on Numerical Analysis

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