Nonparametric estimation of the division rate of a size-structured population

We consider the problem of estimating the division rate of a size-structured population in a nonparametric setting. The size of the system evolves according to a transport-fragmentation equation: each individual grows with a given transport rate and splits into two offspring of the same size, following a binary fragmentation process with unknown division rate that depends on its size. In contrast to a deterministic inverse problem approach, as in [B. Perthame and J. P. Zubelli, Inverse Problems, 23 (2007), pp. 1037-1052; M. Doumic, B. Perthame, and J. Zubelli, Inverse Problems, 25 (2009), pp. 1-22], in this paper we take the perspective of statistical inference: our data consists of a large sample of the size of individuals, when the evolution of the system is close to its time-asymptotic behavior, so that it can be related to the eigenproblem of the considered transport-fragmentation equation. By estimating statistically each term of the eigenvalue problem and by suitably inverting a certain linear operator, we are able to construct a more realistic estimator of the division rate that achieves the same optimal error bound as in related deterministic inverse problems. Our procedure relies on kernel methods with automatic bandwidth selection. It is inspired by model selection and recent results of Goldenshluger and Lepski [A. Goldenshluger and O. Lepski, arXiv:0904.1950, 2009; arXiv:1009.1016, 2010].