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

TL;DR

This paper develops a nonparametric statistical method to estimate the division rate in size-structured populations, leveraging eigenproblem-based inference and kernel methods with automatic bandwidth selection.

Contribution

It introduces a novel statistical inference approach for estimating division rates, improving upon deterministic inverse problem methods with a data-driven, kernel-based estimator.

Findings

Achieves optimal error bounds similar to deterministic methods.

Uses kernel methods with automatic bandwidth selection.

Provides a practical estimator based on large sample data.

Abstract

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.