On the consistent separation of scale and variance for Gaussian random fields

TL;DR

This paper establishes conditions under which the scale and variance of Gaussian random fields with Matérn autocovariance can be consistently estimated in high dimensions, revealing a phase transition at dimension four.

Contribution

It provides the first fixed domain asymptotic results for consistent estimation of scale and variance in Gaussian fields with Matérn covariance when dimension exceeds four.

Findings

Consistent estimation of scale and variance is possible for $d>4$.

Estimates of the irregular term coefficients enable separation of scale and variance.

Results extend to general autocovariance functions and highlight the role of dimension and smoothness.

Abstract

We present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Mat\'ern autocovariance in dimension $d>4$. When $d<4$ this is impossible due to the mutual absolute continuity of Mat\'ern Gaussian random fields with different scale and variance (see Zhang \cite{zhang:2004}). Informally, when $d>4$, we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term. These two coefficients can then be used to separate the scale and variance. We extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions. Our results illustrate the interaction between the accuracy of estimation, the smoothness of the random field, the dimension of the observation space, and the number of increments used for estimation. As a corollary, our results establish the orthogonality of Mat\'ern Gaussian random fields with different parameters when $d>4$. The case $d=4$ is still open.