Goodness-of-fit Tests for Heavy-tailed Random Fields

TL;DR

This paper introduces goodness-of-fit tests for max-stable random fields modeling heavy-tailed spatial data, using Fourier-based statistics and bootstrap methods to assess model adequacy in practical environmental applications.

Contribution

It proposes a novel Fourier transform-based test statistic for max-stable fields and a bootstrap procedure to approximate critical values without explicit covariance expressions.

Findings

Simulation studies validate the asymptotic Gaussian distribution.

The method effectively assesses model fit for PM2.5 and temperature data.

Bootstrap procedure provides reliable critical value approximation.

Abstract

We develop goodness-of-fit tests for max-stable random fields, which are used to model heavy-tailed spatial data. The test statistics are constructed based on the Fourier transforms of the indicators of extreme values in the heavy-tailed spatial data, whose asymptotic distribution is a Gaussian random field under a hypothesized max-stable random field. Since the covariance structure of the limiting Gaussian random field lacks an explicit expression, we propose a stationary bootstrap procedure for spatial fields to approximate critical values. Simulation studies confirm the theoretical distributional results, and applications to PM2.5 and temperature data illustrate the practical utility of the proposed method for model assessment.