Whittle estimation based on the extremal spectral density of a heavy-tailed random field

TL;DR

This paper introduces a novel spectral analysis method for heavy-tailed random fields using extremal spectral density and Whittle estimation, focusing on the largest values in the field.

Contribution

It develops the extremal spectral density and extremal periodogram for heavy-tailed random fields, enabling Whittle estimation for models like Brown-Resnick and max-moving averages.

Findings

Defined the extremal spectral density for heavy-tailed fields.

Proposed the extremal periodogram as an estimator.

Applied Whittle estimation to specific classes of random fields.

Abstract

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random field and regularly varying max-moving averages.