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An important problem in extreme-value theory is the estimation of theprobability that a high-dimensional random vector falls into a given extremefailure set. This paper provides a parametric approach to this problem, basedon a generalization of the tail pairwise dependence matrix (TPDM). The TPDMgives a partial summary of tail dependence for all pairs of components of therandom vector. We propose an algorithm to obtain an approximate completelypositive decomposition of the TPDM. The decomposition is easy to compute andapplicable to moderate to high dimensions. Based on the decomposition, weobtain parameters estimates of a max-linear model whose TPDM is equal to thatof the original random vector. We apply the proposed decomposition algorithm toindustry portfolio returns and maximal wind speeds to illustrate itsapplicability.

Estimating probabilities of multivariate failure sets based on pairwise tail dependence coefficients