Tail Halos: The Covariate Dual of the Tail

2026-09-30

AI Summary

This study addresses the challenge of identifying covariate regions associated with extreme events and high exceedance probabilities. To this end, it introduces the concept of risk and favorable tail halos along with their induced densities, establishing a Bayesian structured additive distributional regression framework. By integrating an extended generalized Pareto distribution with covariate-dependent copulas, the proposed approach enables statistical inference for multidimensional tail regions. Simulation experiments demonstrate that the model accurately recovers nonlinear effects. Furthermore, an application to Edinburgh pollution data successfully reveals specific environmental configurations that trigger extreme events, thereby offering a novel paradigm for extreme risk attribution.

Abstract

Extreme events are often studied through regression models describing how covariates modify tail parameters. In many applications, however, interest lies directly in identifying the parts of the covariate space associated with tail events and with elevated exceedance probabilities. We formalize these ideas through tail halos, covariate-defined regions associated with extremes. Risk halos, when they exist, are minimum-mass covariate regions accounting for a prescribed fraction of exceedance probability, while favorable halos collect covariate values where conditional exceedance probability exceeds its marginal level. To make these set-valued objects interpretable beyond low dimensions, we introduce halo-induced covariate laws and, when densities exist, corresponding halo densities. We develop a Bayesian structured additive distributional regression framework based on extended generalized Pareto marginals, covariate-dependent copulas, and spike-and-slab effect selection to learn marginal and joint halo-induced covariate laws while propagating posterior uncertainty. The approach extends Bayesian shrinkage methods for conditional tail modeling to multivariate bulk-and-tail modeling, nonlinear effect selection, and covariate-region inference. Simulations show recovery of relevant nonlinear effects and increasingly accurate finite-sample halo representations as sample size grows. An application to PM2.5 and NO2 extremes in Edinburgh reveals distinct environmental and temporal configurations associated with high pollution.

Problem

• extreme events
• tail halos
• covariate space
• exceedance probability
• distributional regression

Innovation

• Tail Halos
• Bayesian structured additive distributional regression
• Spike-and-slab effect selection
• Covariate-dependent copulas
• Extreme events