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Multivariate regular variation and the Hüssler-Reiss Pareto model.

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Olivier Zhen Wai

Extreme value theory is a subject with a wide range of applications. For example, in hydrology, we may be interested in estimating the 1000 year return level of a river.Automatic word wrap
One important tool in extreme value theory is the theory of regular variation. Characterisations of the max-domain of attraction are well known in both the univariate and the multivariate cases and rely on regular variation. However, applications drive the development of statistical models to be fitted to data. Therefore, we are interested in developping simple models for multivariate regularly varying random vectors. We first introduce the mathematical framework of the multivariate Breiman lemma. It allows us to recover in a natural way well known models in multivariate extreme value theory. We then focus on a peculiar model : the Hüssler-Reiss Pareto model. Importantly, it fits the theoretical framework of exponential families. We provide also results on the existence, unicity and asymptotic normality of the maximum likelihood estimator.