Testing the Regular Variation Model for Multivariate Extremes with Flexible Circular and Spherical Distributions

TL;DR

This paper introduces flexible probabilistic models for the spectral measure in multivariate extreme value theory, enabling statistical testing of dependence structures using circular and spherical distributions, and applies this to financial data.

Contribution

It proposes a novel approach using nonnegative trigonometric sums for the spectral measure, allowing statistical inference in multivariate extremes.

Findings

Flexible models fit real financial data well.

Statistical tests can assess dependence among extremes.

Enhanced understanding of extremal dependence structures.

Abstract

The regular variation model for multivariate extremes decomposes the joint distribution of the extremes in polar coordinates in terms of the angles and the norm of the random vector as the product of two independent densities: the angular (spectral) measure and the density of the norm. The support of the angular measure is the surface of a unit hypersphere and the density of the norm corresponds to a Pareto density. The dependence structure is determined by the angular measure on the hypersphere, and directions with high probability characterize the dependence structure among the elements of the random vector of extreme values. Previous applications of the regular variation model have not considered a probabilistic model for the angular density and no statistical tests were applied. In this paper, circular and spherical distributions based on nonnegative trigonometric sums are considered flexible probabilistic models for the spectral measure that allows the application of statistical tests to make inferences about the dependence structure among extreme values. The proposed methodology is applied to real datasets from finance.