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 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.
Highlights
The estimation of future extreme episodes of a real process, such as heavyrainfall, heat-waves or simultaneous losses in the financial market, is of crucial importance for risk management
The probabilistic modelling concerns the joint distribution of the random vector of so-called component-wise maxima, in short sample maxima, whose joint distribution is named a multivariate extreme value distribution
Our present proposal has the following key features that make it different from Marcon et al (2015). The use of this particular polynomial expansion makes it possible to accommodate different representations of the dependence structure, such as the Pickands dependence function and the so-called angular measure. This ensures that in each case, there is the fulfillment of some specific constraints which guarantee that a proper extreme value distribution is defined
Summary
The estimation of future extreme episodes of a real process, such as heavyrainfall, heat-waves or simultaneous losses in the financial market, is of crucial importance for risk management. The probabilistic modelling concerns the joint distribution of the random vector of so-called component-wise (block) maxima, in short sample maxima, whose joint distribution is named a multivariate extreme value distribution (de Haan and Ferreira, 2006, Ch. 6) Within this approach, parametric models for the dependence structure have been widely discussed and applied in the literature The use of this particular polynomial expansion makes it possible to accommodate different representations of the dependence structure, such as the Pickands dependence function and the so-called angular (or spectral) measure. This ensures that in each case, there is the fulfillment of some specific constraints which guarantee that a proper extreme value distribution is defined.
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