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

TL;DR

This paper introduces a theoretically justified, computationally efficient Kullback-Leibler divergence-based test for comparing extremal dependence structures in multivariate data, with applications in environmental sciences.

Contribution

It combines extreme value theory with Kullback-Leibler divergence to develop a new test for multivariate extremal dependence, providing theoretical guarantees and practical applicability.

Findings

The test is fast to compute and easy to interpret.

Simulations demonstrate high power of the test.

Applied case study shows seasonal effects on rainfall dependence.

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.