High dimensional inference for extreme value indices

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.

💡 Research Summary

The paper addresses a fundamental problem in multivariate extreme‑value analysis: testing the hypothesis that all marginal distributions share the same extreme‑value index (γ). While a Wald‑type test based on Hill estimators works well in low dimensions, its performance collapses as the number of variables p grows, because the effective sample size for tail estimation (the number of exceedances k) is much smaller than the total sample size n. To overcome this “dimensionality curse,” the authors propose two novel testing procedures that remain valid in high‑dimensional settings and under both weak and general cross‑sectional tail dependence.

The first procedure, called the Gumbel test, constructs a test statistic
(T(k_1,\dots,k_p)=\max_{1\le j\le p}\sqrt{k_j},| \hat\gamma_j(k_j)/\gamma_{0j}-1|),
where (\hat\gamma_j(k_j)) is the Hill estimator for the j‑th margin based on the top (k_j) order statistics and (\gamma_{0j}) are pre‑specified null values. Under three sets of regularity conditions—(A) a uniform second‑order tail condition, (B) a sparsity condition on the covariance matrix of a transformed vector Y that captures tail dependence, and (C) growth restrictions linking p, k_min, and k_max—the authors prove that the transformed statistic (T^2-2\log p+\log\log p) converges in distribution to a Gumbel law (Theorem 1). Consequently, the (1−α) quantile of the limiting Gumbel distribution yields an explicit critical value (c_\alpha=\sqrt{2\log p-\log\log p+q_\alpha}). The sparsity condition (B) is analogous to those used in high‑dimensional mean tests and essentially requires that pairwise tail dependence coefficients are not too large.

The second procedure relaxes the sparsity requirement by employing a multiplier bootstrap. Independent standard normal multipliers (\xi_i) are generated and applied to the original observations to form a bootstrap version of the test statistic, (T_B(k_1,\dots,k_p)). Under a slightly stronger set of conditions (A) and (C′) – which still allow p to diverge but demand a larger effective sample size (e.g., (k_{\min}\log^7 p\to\infty)) – the conditional (1−α) quantile of the bootstrap distribution consistently estimates the true critical value (Theorem 2). This bootstrap approach works even when the tail dependence structure is dense, making it suitable for applications such as climate extremes or systemic financial risk where strong cross‑sectional tail dependence is common.

The authors conduct extensive Monte‑Carlo experiments. They vary p (from modest to several hundred), the proportion of exceedances k, and the strength of tail dependence (both sparse and dense). Results show that both the Gumbel and bootstrap tests maintain nominal size while achieving substantially higher power than the traditional Wald test, especially as p grows. In dense dependence scenarios the bootstrap test outperforms the Gumbel test, confirming its robustness to violations of the sparsity assumption.

Two real‑data applications illustrate practical relevance. First, wind‑speed maxima from a dense network of Dutch weather stations are analyzed; the proposed tests reject the equal‑γ hypothesis, suggesting that spatial extreme‑value models assuming a common tail index may be misspecified. Second, daily returns of S&P 500 constituents are examined; again, the null of identical extreme‑value indices across stocks is rejected, implying that multivariate regular variation models used in portfolio risk assessment should allow heterogeneous tail indices.

In summary, the paper makes three major contributions: (1) it derives the high‑dimensional asymptotic distribution of a maximum‑type statistic built from Hill estimators, establishing a Gumbel limit under mild sparsity; (2) it introduces a multiplier bootstrap that bypasses the sparsity requirement while preserving asymptotic validity; (3) it provides thorough theoretical, simulation, and empirical evidence that the new methods dramatically improve testing of equal extreme‑value indices in high dimensions. These results open the door to more reliable inference in multivariate extreme‑value modeling, spatial extremes, and high‑dimensional risk management.