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Keywords:

  • Bayes;
  • Bivariate extreme value distribution;
  • Extreme conditional quantiles;
  • Markov chain Monte Carlo methods;
  • Metropolis-within-Gibbs sampling;
  • φ-irreducibility;
  • Prediction;
  • Rare event probabilities;
  • Reversible jumps;
  • Spectral measure

Summary.  The tail of a bivariate distribution function in the domain of attraction of a bivariate extreme value distribution may be approximated by that of its extreme value attractor. The extreme value attractor has margins that belong to a three-parameter family and a dependence structure which is characterized by a probability measure on the unit interval with mean equal to inline image, which is called the spectral measure. Inference is done in a Bayesian framework using a censored likelihood approach. A prior distribution is constructed on an infinite dimensional model for this measure, the model being at the same time dense and computationally manageable. A trans-dimensional Markov chain Monte Carlo algorithm is developed and convergence to the posterior distribution is established. In simulations, the Bayes estimator for the spectral measure is shown to compare favourably with frequentist non-parametric estimators. An application to a data set of Danish fire insurance claims is provided.