Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes

TL;DR

This paper demonstrates that maximum likelihood and cross validation estimators for covariance parameters are consistent and asymptotically normal for a class of non-Gaussian processes derived from Gaussian processes, showing robustness to non-Gaussianity.

Contribution

It establishes the asymptotic properties of these estimators for transformed Gaussian processes, a largely unexplored area, and introduces new technical results on covariance matrix decay and quadratic forms.

Findings

Estimators are consistent and asymptotically normal for non-Gaussian processes.

Establishes decay rates of inverse covariance matrix coefficients.

Provides a central limit theorem for quadratic forms of transformed Gaussian processes.

Abstract

The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.