Eigenfunction martingale estimators for interacting particle systems and their mean field limit

TL;DR

This paper introduces a novel eigenfunction martingale estimator for large interacting particle systems, leveraging the mean field limit's eigenvalues and eigenfunctions, with proven asymptotic properties and demonstrated numerical accuracy.

Contribution

It develops a new estimator based on eigenfunctions of the mean field generator, providing asymptotic unbiasedness, normality, and convergence rates, validated through numerical experiments.

Findings

Estimator is asymptotically unbiased and normal.

Convergence rate towards true parameters established.

Numerical experiments confirm theoretical results even with multiple steady states.

Abstract

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.