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

TL;DR

This paper develops a kernel-based method for nonparametric estimation of the division rate in age-dependent branching processes, achieving optimal convergence rates despite data dependencies and censoring.

Contribution

It introduces an optimal-rate kernel estimator for the division rate in supercritical Bellman-Harris processes under complex data conditions.

Findings

Achieves exponential convergence rate depending on the Malthus parameter and smoothness.

Proves the estimator's rate is minimax optimal.

Highlights limitations of using only data from alive particles at time T.

Abstract

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 $k \geq 2$ children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over $[0,T]$. Asymptotics are taken as $T \rightarrow \infty$; 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, under appropriate ergodicity properties, we construct a kernel-based estimator of $B(x)$ that achieves the rate of convergence $\exp(-\lambda_B \frac{\beta}{2\beta+1}T)$, where $\lambda_B$ is the Malthus parameter and $\beta >0$ is the smoothness of the function $B(x)$ in a vicinity 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.