Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations

TL;DR

This paper introduces truncated-likelihood functions for Gaussian random fields to improve covariance estimation, demonstrating their consistency and asymptotic normality under certain conditions, especially for compactly supported functions.

Contribution

It proposes a new class of likelihood approximations based on covariance function approximations, extending asymptotic analysis to generalized Wendland functions and tapering methods.

Findings

Truncated-likelihood estimators are consistent and asymptotically normal for compactly supported covariances.

For non-compactly supported covariances, truncated-tapered likelihood estimators asymptotically minimize KL divergence.

Application to Wendland covariance functions illustrates the theoretical results.

Abstract

Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncated-likelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions. For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.