Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

Clément Albert (1) , Anne Dutfoy (2) , Stéphane Girard (1)

1 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
2 EDF R&D MRI - Management des Risques Industriels

We investigate the asymptotic behavior of the (relative) extrapolation error associated with some estimators of extreme quantiles based on extreme-value theory. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In constrast, the extrapolation error associated with Weissman estimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are then derived and their accuracy is illustrated numerically.

Mots clés: Extrapolation error; Extreme quantiles; Extreme-value theory

Domaines: Statistiques [math.ST]

Clément Albert, Anne Dutfoy, Stéphane Girard. Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles. 2018. ⟨hal-01692544v1⟩