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

Abstract

We model the growth of a cell population by a piecewise deterministic Markov\nbranching tree. Each cell splits into two offsprings at a division rate $B(x)$\nthat depends on its size $x$. The size of each cell grows exponentially in\ntime, at a rate that varies for each individual. We show that the mean\nempirical measure of the model satisfies a growth-fragmentation type equation\nif structured in both size and growth rate as state variables. We construct a\nnonparametric estimator of the division rate $B(x)$ based on the observation of\nthe population over different sampling schemes of size $n$ on the genealogical\ntree. Our estimator nearly achieves the rate $n^{-s/(2s+1)}$ in squared-loss\nerror asymptotically. When the growth rate is assumed to be identical for every\ncell, we retrieve the classical growth-fragmentation model and our estimator\nimproves on the rate $n^{-s/(2s+3)}$ obtained in \\cite{DHRR, DPZ} through\nindirect observation schemes. Our method is consistently tested numerically and\nimplemented on {\\it Escherichia coli} data.\n