Hypothesis testing for tail dependence parameters on the boundary of the parameter space

TL;DR

This paper develops hypothesis tests for tail dependence parameters on the boundary of the parameter space in multivariate extreme-value models, providing asymptotic distributions, simulation evaluations, and real-world financial data applications.

Contribution

It introduces new hypothesis testing methods for boundary parameters in tail dependence models, extending the asymptotic theory and demonstrating practical effectiveness.

Findings

Test statistics perform well in simulations.

Method recovers optimal factors in max-linear models.

Applied to stock indices to characterize dependence.

Abstract

Modelling multivariate tail dependence is one of the key challenges in extreme-value theory. Multivariate extremes are usually characterized using parametric models, some of which have simpler submodels at the boundary of their parameter space. Hypothesis tests are proposed for tail dependence parameters that, under the null hypothesis, are on the boundary of the alternative hypothesis. The asymptotic distribution of the weighted least squares estimator (Einmahl, Kiriliouk and Segers, Extremes 21, pages 205-233, 2018) is given when the true parameter vector is on the boundary of the parameter space, and two test statistics are proposed. The performance of these test statistics is evaluated for the Brown-Resnick model and the max-linear model. In particular, simulations show that it is possible to recover the optimal number of factors for a max-linear model. Finally, the methods are applied to characterize the dependence structure of two major stock market indices, the DAX and the CAC40.