The LAN property for McKean-Vlasov models in a mean-field regime

TL;DR

This paper proves the LAN property for parameter estimation in large systems of interacting particles modeled by McKean-Vlasov equations, providing sharp asymptotic results and criteria for identifiability.

Contribution

It establishes the LAN property for multidimensional parameters in mean-field particle systems, with explicit conditions for Fisher information and asymptotic optimality.

Findings

LAN property proven for mean-field particle models

Explicit criteria for Fisher information non-degeneracy

Asymptotic minimax optimality of estimators

Abstract

We establish the local asymptotic normality (LAN) property for estimating a multidimensional parameter in the drift of a system of $N$ interacting particles observed over a fixed time horizon in a mean-field regime $N \rightarrow \infty$. By implementing the classical theory of Ibragimov and Hasminski, we obtain in particular sharp results for the maximum likelihood estimator that go beyond its simple asymptotic normality thanks to H\'ajek's convolution theorem and strong controls of the likelihood process that yield asymptotic minimax optimality (up to constants). Our structural results shed some light to the accompanying nonlinear McKean-Vlasov experiment, and enable us to derive simple and explicit criteria to obtain identifiability and non-degeneracy of the Fisher information matrix. These conditions are also of interest for other recent studies on the topic of parametric inference for interacting diffusions.