article · 18/01/2015
Statistical estimation of a growth-fragmentation model observed on a genealogical tree
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 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.
Cite this article
Doumic, M., Hoffmann, M., Krell, N., & Robert, L. (2015). Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli, 21(3), 1760-1799. https://doi.org/10.3150/14-BEJ623@article{Doumic2015_303,
author = {Doumic, Marie and Hoffmann, Marc and Krell, Nathalie and Robert, Lydia},
year = {2015},
month = {1},
title = {Statistical estimation of a growth-fragmentation model observed on a genealogical tree},
journal = {Bernoulli},
volume = {21},
number = {3},
pages = {1760-1799},
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 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.},
doi = {10.3150/14-BEJ623},
}TY - JOUR
AU - Doumic, Marie
AU - Hoffmann, Marc
AU - Krell, Nathalie
AU - Robert, Lydia
PY - 2015
DA - 2015/01/18
TI - Statistical estimation of a growth-fragmentation model observed on a genealogical tree
JO - Bernoulli
VL - 21
IS - 3
AB - 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.
SP - 1760
EP - 1799
DO - 10.3150/14-BEJ623
ER -