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Statistical estimation of a growth-fragmentation model observed on a genealogical tree

Doumic, Marie; Hoffmann, Marc; Krell, Nathalie; Robert, Lydia (2015), Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21, 3, p. 1760-1799. 10.3150/14-BEJ623

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Type
Article accepté pour publication ou publié
Date
2015
Journal name
Bernoulli
Volume
21
Number
3
Publisher
International Statistical Institute
Pages
1760-1799
Publication identifier
10.3150/14-BEJ623
Metadata
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Author(s)
Doumic, Marie
Laboratoire Jacques-Louis Lions [LJLL]
Hoffmann, Marc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Krell, Nathalie cc
Institut de Recherche Mathématique de Rennes [IRMAR]
Robert, Lydia
MICrobiologie de l'ALImentation au Service de la Santé [MICALIS]
Abstract (EN)
We raise the issue of estimating the division rate for a growing and dividing population modelled by a piecewise deterministic Markov branching tree. Such models have broad applications, ranging from TCP/IP window size protocol to bacterial growth. Here, the individ-uals split into two offsprings at a division rate B(x) that depends on their size x, whereas their size grow exponentially in time, at a rate that exhibits variability. The mean empirical measure of the model satisfies a growth-fragmentation type equation, and we bridge the determinis-tic and probabilistic viewpoints. We then construct a nonparametric estimator of the division rate B(x) based on the observation of the pop-ulation over different sampling schemes of size n on the genealogical tree. Our estimator nearly achieves the rate n −s/(2s+1) in squared-loss error asymptotically, generalizing and improving on the rate n −s/(2s+3) obtained in [13, 15] through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data, which demonstrates its major interest for practical applications.
Subjects / Keywords
Nonparametric estimation; Growth-fragmentation; cell division equation; Markov chain on a tree

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