Bayesian Inference for the Extremal Dependence

TL;DR

This paper introduces a nonparametric Bayesian method using Bernstein polynomials to estimate extremal dependence in multivariate extremes, enabling flexible inference of dependence functions with valid constraints.

Contribution

It develops a fully nonparametric Bayesian approach for extremal dependence estimation using polynomial priors, addressing constraints via coefficient priors and employing trans-dimensional MCMC.

Findings

Efficient inference demonstrated through simulation studies.

Applied to exchange rate data revealing extremal dependence patterns.

Method provides a flexible framework for modeling multivariate extremes.

Abstract

A simple approach for modeling multivariate extremes is to consider the vector of component-wise maxima and their max-stable distributions. The extremal dependence can be inferred by estimating the angular measure or, alternatively, the Pickands dependence function. We propose a nonparametric Bayesian model that allows, in the bivariate case, the simultaneous estimation of both functional representations through the use of polynomials in the Bernstein form. The constraints required to provide a valid extremal dependence are addressed in a straightforward manner, by placing a prior on the coefficients of the Bernstein polynomials which gives probability one to the set of valid functions. The prior is extended to the polynomial degree, making our approach fully nonparametric. Although the analytical expression of the posterior is unknown, inference is possible via a trans-dimensional MCMC scheme. We show the efficiency of the proposed methodology by means of a simulation study. The extremal behaviour of log-returns of daily exchange rates between the Pound Sterling vs the U.S. Dollar and the Pound Sterling vs the Japanese Yen is analysed for illustrative purposes.