Volume 21, 2017
|Page(s)||275 - 302|
|Published online||12 December 2017|
Estimating the division kernel of a size-structured population
Laboratoire Paul Painlevé UMR CNRS 8524, Université de Lille 1, France.
Received: 3 April 2017
Revised: 14 June 2017
Accepted: 14 June 2017
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 α. At division, the cells divide at a constant rate R and share their content between the two daughter cells into fractions Γ and 1 − Γ where Γ 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.
Mathematics Subject Classification: 60J80 / 60J85 / 62G05 / 62G07 / 92D25
Key words: Random size-structured population / division kernel / nonparametric estimation / Goldenshluger-Lepski’s method / adaptive estimator / penalization
© EDP Sciences, SMAI, 2017
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