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Hypothesis testing for partial tail correlation in multivariate extremes

Statistical modeling of high dimensional extremes remains challenging and has generally been limited to moderate dimensions. Understanding structural relationships among variables at their extreme levels is crucial both for constructing simplified models and for identifying sparsity in extremal dependence. In this paper, we introduce the notion of partial tail correlation to characterize structural relationships between pairs of variables in their tails. To this end, we propose a tail regression approach for nonnegative regularly varying random vectors and define partial tail correlation based on the regression residuals. Using an extreme analogue of the covariance matrix, we show that the resulting regression coefficients and partial tail correlations take the same form as in classical non-extreme settings. For inference, we develop a hypothesis test to explore sparsity in extremal dependence structures, and demonstrate its effectiveness through simulations and an application to the Danube river network.