Nonparametric estimation of covariance functions by model selection

TL;DR

This paper introduces a model selection method for estimating covariance functions of multi-dimensional stochastic processes, providing a non-asymptotic analysis and an oracle inequality to ensure optimal model choice.

Contribution

It presents a novel nonparametric covariance estimation approach using basis function expansion and a data-driven model selection procedure with theoretical guarantees.

Findings

The estimator performs well under general assumptions.

The model selection procedure is proven to be optimal via oracle inequality.

The approach is applicable to i.i.d. replications of stochastic processes.

Abstract

We propose a model selection approach for covariance estimation of a multi-dimensional stochastic process. Under very general assumptions, observing i.i.d replications of the process at fixed observation points, we construct an estimator of the covariance function by expanding the process onto a collection of basis functions. We study the non asymptotic property of this estimate and give a tractable way of selecting the best estimator among a possible set of candidates. The optimality of the procedure is proved via an oracle inequality which warrants that the best model is selected.