Multivariate peaks-over-threshold with latent variable representations of Generalized Pareto vectors

Generalized Pareto distributions with a positive tail index arise from allowing the rate of an exponential variable to vary according to a Gamma distribution. In this paper, we exploit this property to define a flexible and statistically tractable modeling framework for multivariate extremes based on componentwise ratios of between any two random vectors with exponential and Gamma marginal distributions. To model multivariate threshold exceedances, we propose hierarchical constructions using a latent random vector with Gamma margins, whose Laplace transform is key to obtaining the multivariate distribution function. We refer to these new models as Gamma-Mixture Generalized Pareto Distributions (GMGPD). The extremal dependence properties of such constructions, covering asymptotic independence and asymptotic dependence, are studied. We detail two useful parametric model classes: the latent Gamma vectors are sums of independent Gamma components in the first construction (called the convolution model), whereas they correspond to chi-squared random vectors in the second construction. Both of these constructions exhibit asymptotic independence, and we further propose a parametric extension (called Beta-scaling) to obtain asymptotic dependence. We demonstrate satisfactory performance of pairwise-likelihood inference for several scenarios through a simulation study for trivari- ate models, including a hybrid model mixing bivariate subvectors with asymptotic dependence and independence. An application to multivariate air pollution data illustrates the flexibility of the new class of models.