Estimating the division kernel of a size-structured population

Hoang, Van Ha

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 275-302.

We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rate $\alpha$. At division, the cells divide at a constant rate $R$ and share their content between the two daughter cells into fractions $\Gamma$ and $1-\Gamma$ where $\Gamma$ has a symmetric density $h$ on $[0,1]$, since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval $[0,T]$ with fixed $T$. We address here the problem of estimating the division kernel $h$ (or fragmentation kernel) when the division tree is completely observed. An adaptive estimator of $h$ is constructed based on a kernel function $K$ with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.

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