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High dimensional inference for extreme value indices

Liujun Chen, Chen Zhou

TL;DR

This work tackles high-dimensional tail inference by introducing Hill-estimator-based tests for equality of marginal extreme value indices across many dimensions. It develops a Gumbel-limit approach for weak dependence and a multiplier bootstrap approach for general dependence, with parallel tests for both $H_0$ and $H_0^*$ and rigorous high-dimensional asymptotics under second-order conditions. Simulation and real-data analyses show that Wald-type tests deteriorate in high dimensions, while the proposed Gumbel and bootstrap procedures achieve reliable size control and competitive power, including under strong tail dependence. The methods enable robust, scalable inference for tail risk in multivariate extremes and provide practical guidance for validating key modeling assumptions in high-dimensional settings.

Abstract

When applying multivariate extreme value statistics to analyze tail risk in compound events defined by a multivariate random vector, one often assumes that all dimensions share the same extreme value index. While such an assumption can be tested using a Wald-type test, the performance of such a test deteriorates as the dimensionality increases. This paper introduces novel tests for comparing extreme value indices in highdimensional settings, under both weak and general cross-sectional tail dependence. We establish the asymptotic behavior of the proposed tests. The proposed tests significantly outperform existing methods in high-dimensional scenarios in simulations. We demonstrate real-life applications of the proposed tests for two datasets previously assumed to have identical extreme value indices across all dimensions.

High dimensional inference for extreme value indices

TL;DR

This work tackles high-dimensional tail inference by introducing Hill-estimator-based tests for equality of marginal extreme value indices across many dimensions. It develops a Gumbel-limit approach for weak dependence and a multiplier bootstrap approach for general dependence, with parallel tests for both and and rigorous high-dimensional asymptotics under second-order conditions. Simulation and real-data analyses show that Wald-type tests deteriorate in high dimensions, while the proposed Gumbel and bootstrap procedures achieve reliable size control and competitive power, including under strong tail dependence. The methods enable robust, scalable inference for tail risk in multivariate extremes and provide practical guidance for validating key modeling assumptions in high-dimensional settings.

Abstract

When applying multivariate extreme value statistics to analyze tail risk in compound events defined by a multivariate random vector, one often assumes that all dimensions share the same extreme value index. While such an assumption can be tested using a Wald-type test, the performance of such a test deteriorates as the dimensionality increases. This paper introduces novel tests for comparing extreme value indices in highdimensional settings, under both weak and general cross-sectional tail dependence. We establish the asymptotic behavior of the proposed tests. The proposed tests significantly outperform existing methods in high-dimensional scenarios in simulations. We demonstrate real-life applications of the proposed tests for two datasets previously assumed to have identical extreme value indices across all dimensions.
Paper Structure (21 sections, 19 theorems, 233 equations, 3 figures, 4 tables)

This paper contains 21 sections, 19 theorems, 233 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Test for $H_0$
  3. Test under weak dependence
  4. Test under general dependence
  5. Power analysis
  6. Test for equal extreme value indices hypothesis $H_0^*$
  7. Test under weak dependence
  8. Test under general dependence
  9. Simulation
  10. Simulation setting
  11. Simulation results
  12. Sensitivity Analysis
  13. Application
  14. Proof of Theorem \ref{['theorem:max:i']}
  15. Preliminary Lemmas
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that Conditions condition:SOC, Eigens and condition:k:choice hold. Under the null hypothesis $H_0$, as $n\to\infty$, for any $x\in \mathbb{R}$,

Figures (3)

  • Figure 1: The rejection rates of the multiplier bootstrap test and the Gumbel test for different values of $c$.
  • Figure 2: Left: the $p$-values of the tests against different levels of $k$ for the wind gust data. The black horizontal line indicates the significance level, $\alpha = 0.05$. Right: the Hill estimates $\widehat{\gamma}_j$ with 95% confidence intervals for the wind gust data.
  • Figure 3: Left: the $p$-values of the tests against different levels of $k$ for the portfolio data. The black horizontal line indicates the significance level, $\alpha = 0.05$. Right: the Hill estimates $\widehat{\gamma}_j$ with 95% confidence intervals for the portfolio data.

Theorems & Definitions (35)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • Theorem 6
  • Lemma S1
  • Lemma S2
  • ...and 25 more