Simple models for multivariate regular variation and the Hüsler-Reiß Pareto distribution. (English) Zbl 1473.62170
Summary: 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.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62G32 | Statistics of extreme values; tail inference |
| 62E15 | Exact distribution theory in statistics |
Keywords:
exponential family; extreme value theory; maximum likelihood estimation; regular variationsSoftware:
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