Stein estimation for the drift of Gaussian processes using the Malliavin calculus

TL;DR

This paper develops Stein-type superefficient estimators for the drift of Gaussian processes using Malliavin calculus, extending classical James-Stein estimators to a nonparametric functional setting.

Contribution

It introduces a novel approach to construct Stein-type estimators for Gaussian process drifts via Malliavin calculus and superharmonic functionals, advancing nonparametric estimation methods.

Findings

Constructed superefficient estimators using Malliavin calculus

Extended James--Stein estimators to Gaussian processes

Validated estimators through numerical simulations

Abstract

We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].