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.

中文翻译：

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与极端分位数估计相关的外推误差的渐近行为

我们研究了基于极值理论的与极端分位数的一些估计量相关的（相对）外推误差的渐近行为。结果表明，外推误差可以解释为一阶泰勒展开式的余数。然后提供必要和充分的条件，使得随着样本量的增加，该误差趋于零。有趣的是，在所谓的指数尾估计器的情况下，这些条件导致将Gumbel最大吸引域细分为三个子集。相反，与Weissman估计量相关的外推误差在整个Fréchet最大吸引域上具有共同的行为。然后导出外推误差的一阶等价物，表明在某些Gumbel最大吸引域子集上，Weissman估计量可能导致比指数尾估计量小的外推误差。用数字方式说明了等效项的准确性，并且还提供了对实际数据的应用。