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

TL;DR

The paper introduces a Kullback–Leibler divergence-based two-sample test for extremal dependence in multivariate data, grounded in multivariate regular variation and operationalized via a homogeneous risk functional $r$ and a fixed partition of the exceedance region. The statistic $\hat{D}_K$ quantifies regionwise differences in extreme-event probabilities $p_j$ and $q_j$, with asymptotic $\chi^2(K-1)$ behavior under known marginals and bootstrap-based critical values when marginals are unknown. Simulations demonstrate favorable size and power properties across a range of risk-function choices, partition schemes, and margin knowledge, and an application to French rainfall shows seasonal shifts in extremal dependence between 6-minute and hourly precipitation aggregations. The approach is fast to compute, interpretable, and broadly applicable to climate, hydrology, and risk management contexts where extremal dependence structure matters.

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.

Figures (9)

• Figure 1: Left: scatter plots of daily maxima of hourly and 6-minute rainfall recorded in Bordeaux (France) in winter (top) and spring (bottom) during the period 2006--2023. Right: extremal correlation coefficient $\chi(v)$ with confidence intervals for this data, measuring the strength of dependence above the quantile level $v$; see also \ref{['eq: chi']}.
• Figure 2: Three risk functionals $r(\boldsymbol X)$ with partition examples obtained on simulated bivariate draws of $\boldsymbol X_1,\dots, \boldsymbol X_n$. Light gray dots are non extreme realizations and circled points in each panel represents extremes, i.e., $\{ \boldsymbol X_i: r(\boldsymbol X_i) >u \}$, defined with respect to the risk functional $r(\boldsymbol x) = \max(x_1,x_2)$ (left panel); $r(\boldsymbol x) = \min(x_1,x_2)$ (center panel); $r(\boldsymbol x) = \sqrt{x_1^2 + x_2^2}$ (right panel). The region $\{ r(\boldsymbol X) >u \}$ is divided in $K$ subsets in order to estimate \ref{['cond_prob']} with $K=3, 4$ and $5$ for the right, middle and right panel, respectively.
• Figure 3: Histograms of bootstrap distribution (blue) and true null distribution (pink) together with the density of a chi-squared distribution with $K-1=3$ degrees of freedom, for the situation when the margins are known (left), and when the margins are unknown and have to be normalized empirically (right).
• Figure 4: Mean (solid blue line) and empirical $5\%$ and $95\%$ quantiles (dashed blue lines) of 500 samples of KL test statistic $\hat{D}_K$ based on Euclidean risk functional with $K=5$ sets for a range of exceedances $k_n$ together with the critical values (black dashed line) at level $95\%$ and rejection percentages (orange line). Top and bottom rows show results for known and unknown margins, respectively. The two samples are generated from the same distribution (left), from distributions with different extremal dependence structure (center), and from different distributions with same extremal dependence structure (right).
• Figure 5: Rejection percentages of $H_0$ for 500 simulations of Clayton vs Clayton (left) and Dirichlet vs Clayton (right) for different choices of risk functionals: the maximum risk (dotted line), the Euclidean norm (solid line) and the sum risk (dashed line). Blue shaded area indicates significance level.

Theorems & Definitions (19)

• Remark 1
• Example 1
• Example 2
• Example 3
• Proposition 1
• Theorem 2
• Theorem 3
• Example 4
• Example 5
• Example 6