Nonparametric adaptive estimation for interacting particle systems

Abstract

AbstractWe consider a stochastic system of interacting particles with constant diffusion coefficient and drift linear in space, time‐depending on two unknown deterministic functions. Our concern here is the nonparametric estimation of these functions from a continuous observation of the process on for fixed and large . We define two collections of projection estimators belonging to finite‐dimensional subspaces of . We study the ‐risks of these estimators, where the risk is defined either by the expectation of an empirical norm or by the expectation of a deterministic norm. Afterwards, we propose a data‐driven choice of the dimensions and study the risk of the adaptive estimators. The results are illustrated by numerical experiments on simulated data.