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

2025-12-06

AI Summary

Existing goodness-of-fit (GOF) tests for max-stable random fields (MSRFs) modeling heavy-tailed spatial extremal data lack theoretical validity and practical feasibility. Method: We propose a novel GOF test based on the Fourier transform of the extremal indicator function; its test statistic converges weakly to a Gaussian random field. To address the intractability of the limiting covariance structure—lacking a closed-form expression—we develop a stationary bootstrap resampling scheme tailored to spatial data, enabling accurate approximation of the asymptotic covariance field. Results: Simulation studies and real-data applications (PM₂.₅ and temperature extremes) demonstrate that the method substantially improves reliability and robustness under complex spatial dependence. It constitutes the first rigorous, implementable GOF procedure for heavy-tailed spatial extreme-value models.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Develop goodness-of-fit tests for max-stable random fields
• Model heavy-tailed spatial data using extreme value indicators
• Propose bootstrap for critical values in spatial model assessment

Innovation

• Fourier transforms of extreme value indicators
• Stationary bootstrap for spatial fields
• Testing max-stable models for heavy-tailed data