  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 offsprings 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 (Perthame,
Zubelli, 2007) and (Doumic, Perthame, Zubelli, 2009), we take in this paper the
perspective of statistical inference: our data consists in 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 (see \cite{PR} for instance). By
estimating statistically each term of the eigenvalue problem and by suitably
inverting a certain linear operator (see previously quoted articles), 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 Goldenschluger and Lepski.
