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We revisit multivariate extreme value theory modeling by emphasizingmultivariate regular variations and the multivariate Breiman Lemma. This allowsus to recover in a simple framework the most popular multivariate extreme valuedistributions, such as the logistic, negative logistic, Dirichlet, extremal-$t$and H\"usler-Reiss models. In a second part of the paper, we focus on theH\"usler-Reiss Pareto model and its surprising exponential family property.After a thorough study of this exponential family structure, we focus onmaximum likelihood estimation. We also consider the generalized H\"usler-ReissPareto model with different tail indices and a likelihood ratio test fordiscriminating constant tail index versus varying tail indices.

Simple models for multivariate regular variations and the H\"usler-Reiss Pareto distribution