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

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.

References in corpus (4)

• Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality
• Eigenelements of a General Aggregation-Fragmentation Model
• On the Inverse Problem for a Size-Structured Population Model
• Uniform bounds for norms of sums of independent random functions

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• Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation
• Cell lineage statistics with incomplete population trees
• Minimal penalty for Goldenshluger-Lepski method
• Adaptive nonparametric estimation of a component density in a two-class mixture model
• A convergent numerical algorithm for the stochastic growth-fragmentation problem