Bernoulli

  • Bernoulli
  • Volume 21, Number 3 (2015), 1760-1799.

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

Marie Doumic, Marc Hoffmann, Nathalie Krell, and Lydia Robert

Full-text: Open access

Abstract

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 individuals 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 deterministic and probabilistic viewpoints. We then construct a nonparametric estimator of the division rate $B(x)$ based on the observation of the population 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 ( SIAM J. Numer. Anal. 50 (2012) 925–950, Inverse Problems 25 (2009) 1–22) through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data, which demonstrates its major interest for practical applications.

Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1760-1799.

Dates
Received: September 2013
Revised: March 2014
First available in Project Euclid: 27 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1432732036

Digital Object Identifier
doi:10.3150/14-BEJ623

Mathematical Reviews number (MathSciNet)
MR3352060

Zentralblatt MATH identifier
06470456

Keywords
cell division equation growth-fragmentation Markov chain on a tree nonparametric estimation

Citation

Doumic, Marie; Hoffmann, Marc; Krell, Nathalie; Robert, Lydia. Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli 21 (2015), no. 3, 1760--1799. doi:10.3150/14-BEJ623. https://projecteuclid.org/euclid.bj/1432732036


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