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 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.