On the maximum likelihood estimator for the generalized extreme-value distribution
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Date
2016
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Abstract
The vanilla method in univariate extreme-value theory consists of fitting
the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of
block maxima. Despite claims to the contrary, the asymptotic normality of the maximum
likelihood estimator has never been established. In this paper, a formal proof
is given using a general result on the maximum likelihood estimator for parametric
families that are differentiable in quadratic mean but whose support depends on the
parameter. An interesting side result concerns the (lack of) differentiability in quadratic
mean of the GEV family.
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Keywords
differentiability in quadratic mean, support, Lipschitz condition, generalized extreme-value distribution, Fisher information, empirical process, maximum likelihood, M-estimator