Statistical estimation in a randomly structured branching population

TL;DR

This paper develops statistical methods for estimating parameters and transition functions in a complex branching process where traits evolve via diffusion, achieving optimality in large-sample limits.

Contribution

It introduces nonparametric and parametric estimators for a stochastic branching process with trait evolution, demonstrating their asymptotic efficiency and optimality.

Findings

Asymptotic efficiency of parametric estimators

Minimax optimality of nonparametric estimators

Consistent estimation based on first n generations

Abstract

We consider a binary branching process structured by a stochastic trait that evolves according to a diffusion process that triggers the branching events, in the spirit of Kimmel's model of cell division with parasite infection. Based on the observation of the trait at birth of the first n generations of the process, we construct nonparametric estimator of the transition of the associated bifurcating chain and study the parametric estimation of the branching rate. In the limit, as n tends to infinity, we obtain asymptotic efficiency in the parametric case and minimax optimality in the nonparametric case.