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

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.