Bayesian inference for the extremal dependence

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.MSC 2010 subject classifications: 62G05, 62G07, 62G32.

Estimation and uncertainty quantification for extreme quantile regions

“…where z = (z 1 , z 2 ) with z i = z i (y i ; θ) given by (3.4) and where ϑ = (θ, h). Following Marcon et al (2016) we model the angular density using Bernstein polynomials. Writing L(z) = (z 1 + z 2 )A(v) with v = z 2 /(z 1 + z 2 ), we model Pickands dependence function A(v) through a Bernstein polynomial of degree κ = 0, 1, .…”

“…. , β κ ) is a parameter vector satisfying suitable conditions so that A κ in (3.6) defines a valid Pickands dependence function (Marcon et al, 2016, Section 3.1), and Be(·; a, b) is the beta density function with parameters a > 0 and b > 0. Modelling Pickands dependence function with a polynomial in Bernstein form is equivalent to modelling the angular distribution with a Bernstein polynomial.…”

“…We infer the marginal GEV parameters and the extremal dependence structure through a semiparametric Bayesian approach. In particular, we combine the inferential scheme for each marginal parameter set θ 1 , θ 2 described above with the transdimensional MCMC scheme for inferring the dependence structure over the unknown number of elements in β κ suggested by Marcon et al (2016) (see also Antoniano-Villalobos and Walker 2013;Fan and Sisson 2011). The latter takes into account that at each MCMC iteration the dimension of β κ changes with κ and that the size of β κ is potentially infinite (see Marcon et al, 2016, Section 3.2 for details).…”

“…Inference is performed using an adaptive random-walk Metropolis-Hastings algorithm (Garthwaite et al, 2016). In the bivariate case we combine the univariate approach for the estimation of the GEV marginal parameters with the non-parametric Bayesian approach for the estimation of the extremal dependence proposed by Marcon et al (2016). As a result we obtain a semi-parametric Bayesian inferential method based on the censoredlikelihood corresponding to a suitable two-dimensional extreme-value distribution.…”

Regression‐type models for extremal dependence

“…Therefore, most literature has focused on the estimation of the extremal dependence structures described by spectral measures or equivalently angular densities (Boldi & Davison, 2007;de Carvalho, Oumow, Segers, & Warchoł, 2013;Einmahl, Li, & Liu, 2009;Hanson, de Carvalho, & Chen, 2017;Sabourin & Naveau, 2014). Related quantities, such as the Pickands dependence function (Pickands, 1981) and the stable tail dependence function (Drees & Kaufmann, 1998;Huang, 1992), were investigated by many authors (Einmahl, de Haan, & Li, 2006;Gudendorf & Segers, 2012;Marcon, Padoan, & Antoniano-Villalobos, 2016;Wadsworth & Tawn, 2013). A wide variety of parametric models for the spectral density that allow flexible dependence structures were proposed (Kotz & Nadarajah, 2000, sec.…”