Nonparametric Estimation from Correlated Copies of a Drifted Process

TL;DR

This paper develops nonparametric estimation methods for drift functions and their derivatives from multiple correlated process copies, providing risk bounds and model selection guarantees for Gaussian processes.

Contribution

It introduces non-asymptotic risk bounds for drift and derivative estimators from correlated samples, with improved bounds and model selection procedures for Gaussian processes.

Findings

Risk bounds for drift function estimators

Sharper bounds for derivative estimators in Gaussian cases

Model selection procedures with theoretical guarantees

Abstract

This paper presents several situations leading to the observation of multiple correlated copies of a drifted process, and then non-asymptotic risk bounds are established on nonparametric estimators of the drift function $b_0$ and its derivative. For drifted Gaussian processes with a regular enough covariance function, a sharper risk bound is established on the estimator of $b_0'$, and a model selection procedure is provided with theoretical guarantees.