Nonparametric estimation of the division rate of an age dependent branching process.

Summary We study the nonparametric estimation of the branching rate B(x) of a supercritical Bellman-Harris population: a particle with age x has a random lifetime governed by B(x). at its death time, it gives rise to k ≥ 2 offsprings with lifetime governed by the same division rate and so on. We observe continuously the process over a large time interval [0, T ]. the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, we construct a kernel-based estimator of B(x) that achieves the rate of convergence exp(−λ B β 2β+1 T), where λ B is the Malthus parameter and β > 0 is the smoothness of the function B(x) in a viscinity of x. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time T only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.