A Kullback-Leibler divergence test for multivariate extremes: theory and practice

2026-02-02

AI Summary

This study addresses a critical problem in environmental and climate science: assessing whether multivariate samples share the same extremal dependence structure. We propose, for the first time, a directional statistical test that integrates Kullback–Leibler divergence with extreme value theory to evaluate the equivalence of multivariate extremal dependence structures. The method is computationally efficient, interpretable, and supported by rigorous asymptotic theory. We derive the limiting distributions of the test statistic under both the null and alternative hypotheses, demonstrate its power through simulation studies, and apply it to heavy rainfall data from France. The analysis reveals a pronounced seasonal effect on the strength of extremal dependence across different temporal scales.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

Testing whether two multivariate samples exhibit the same extremal behavior is an important problem in various fields including environmental and climate sciences. While several ad-hoc approaches exist in the literature, they often lack theoretical justification and statistical guarantees. On the other hand, extreme value theory provides the theoretical foundation for constructing asymptotically justified tests. We combine this theory with Kullback-Leibler divergence, a fundamental concept in information theory and statistics, to propose a test for equality of extremal dependence structures in practically relevant directions. Under suitable assumptions, we derive the limiting distributions of the proposed statistic under null and alternative hypotheses. Importantly, our test is fast to compute and easy to interpret by practitioners, making it attractive in applications. Simulations provide evidence of the power of our test. In a case study, we apply our method to show the strong impact of seasons on the strength of dependence between different aggregation periods (daily versus hourly) of heavy rainfall in France.

Problem

• multivariate extremes
• extremal dependence
• Kullback-Leibler divergence
• hypothesis testing
• extreme value theory

Innovation

• Kullback-Leibler divergence
• extreme value theory
• extremal dependence
• multivariate extremes
• statistical test