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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 contrast, 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 showing that Weissman estimator may lead to smaller extrapolation errors than the Exponential Tail estimator on some subsets of Gumbel maximum domain of attraction. The accuracy of the equivalents is illustrated numerically and an application on real data is also provided.

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