Asymptotic analysis of maximum likelihood estimation of covariance parameters for Gaussian processes: an introduction with proofs

TL;DR

This paper introduces the asymptotic behavior of maximum likelihood estimators for covariance parameters in Gaussian processes, covering both increasing- and fixed-domain settings with proofs and key results.

Contribution

It provides an accessible overview of existing asymptotic results and proof techniques for covariance parameter estimation in Gaussian processes.

Findings

All covariance components are consistently estimable under increasing-domain asymptotics.

Only microergodic parameters are consistently estimable under fixed-domain asymptotics.

For isotropic Matérn functions, only a combination of variance and scale is microergodic.

Abstract

This article provides an introduction to the asymptotic analysis of covariance parameter estimation for Gaussian processes. Maximum likelihood estimation is considered. The aim of this introduction is to be accessible to a wide audience and to present some existing results and proof techniques from the literature. The increasing-domain and fixed-domain asymptotic settings are considered. Under increasing-domain asymptotics, it is shown that in general all the components of the covariance parameter can be estimated consistently by maximum likelihood and that asymptotic normality holds. In contrast, under fixed-domain asymptotics, only some components of the covariance parameter, constituting the microergodic parameter, can be estimated consistently. Under fixed-domain asymptotics, the special case of the family of isotropic Mat\'ern covariance functions is considered. It is shown that only a combination of the variance and spatial scale parameter is microergodic. A consistency and asymptotic normality proof is sketched for maximum likelihood estimators.