Maximum likelihood estimation of mean functions for Gaussian processes under small noise asymptotics

TL;DR

This paper develops maximum likelihood estimators for Gaussian process mean functions under small noise asymptotics, providing explicit likelihood forms, asymptotic normality, efficiency results, and model selection criteria.

Contribution

It identifies the broadest class of mean functions with explicit likelihood expressions and analyzes their asymptotic properties under small noise conditions.

Findings

Likelihood functions explicitly derived for a wide class of mean functions

Asymptotic normality established under small noise asymptotics

Introduction of efficient M-estimators and model selection criteria

Abstract

Maximum likelihood estimators for time-dependent mean functions within Gaussian processes are provided in the context of continuous observations. We find the widest possible class of mean functions for which the likelihood function can be written explicitly. When it is subjected to a small noise asymptotic condition leading to the vanishing of the primary Gaussian noise, we attain local asymptotic normality results, accompanied by insights into the asymptotic efficiency of these estimators. In addition, we introduce M-estimators based on discrete samples, which also leads us to the asymptotic efficiency. Furthermore, we provide quasi-information criteria for model selection analogous to Akaike Information Criteria in discretely observed cases.