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Nonparametric estimation of the division rate of a size-structured population

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Date
2012
Link to item file
http://hal.archives-ouvertes.fr/hal-00578694/fr/
Dewey
Analyse
Sujet
Aggregation-fragmentation h equations; Adaptation; Oracle inequalities; Cell-division equation; Nonparametric density estimation; Statistical inverse problems; Lepski method
Journal issue
SIAM Journal on Numerical Analysis
Volume
50
Number
2
Publication date
2012
Article pages
925-950
Publisher
SIAM
DOI
http://dx.doi.org/10.1137/110828344
URI
https://basepub.dauphine.fr/handle/123456789/5986
Collections
  • CEREMADE : Publications
Metadata
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Author
Doumic, Marie
1005052 Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Hoffmann, Marc
29 Laboratoire d'Analyse et de Mathématiques Appliquées [LAMA]
2579 Centre de Recherche en Économie et Statistique [CREST]
Reynaud-Bouret, Patricia
26 Laboratoire Jean Alexandre Dieudonné [JAD]
Rivoirard, Vincent
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
We 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.

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