Simple models for multivariate regular variation and the Hüsler–Reiß Pareto distribution

Abstract

We revisit multivariate extreme-value theory modeling by emphasizing multivariate regular variation and a multivariate version of Breiman’s Lemma. This allows us to recover in a simple framework the most popular multivariate extreme-value distributions, such as the logistic, negative logistic, Dirichlet, extremal- t and Hüsler–Reiß models. We then focus on the Hüsler–Reiß Pareto model and its surprising exponential family property. After a thorough study of this exponential family structure, we focus on maximum likelihood estimation: we prove the existence of asymptotically normal maximum likelihood estimators and provide simulation experiments assessing their finite-sample properties. We also consider the generalized Hüsler–Reiß Pareto model with different tail indices and a likelihood ratio test for discriminating constant tail index versus varying tail indices.