Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the n observations of interest another n + m observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. We establish the asymptotic normality of this new estimator. It shows greatly improved behavior relative to the Hill estimator, in particular the asymptotic variance is substantially reduced, whereas we can keep the asymptotic bias the same. A simulation study confirms the substantially improved performance of our adapted estimator. We also present an application to the aforementioned earthquake losses.

中文翻译：

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使用相关变量改进了对极值指数的估计

重尾现象自然可以通过极值统计进行分析。这种分析中的关键步骤是估计极值指数，该指数描述了潜在概率分布的尾部沉重程度。我们考虑这样的情况，在我们感兴趣的n个观测值旁边还有另一个n + m对一个或多个相关变量（例如，由于地震造成的财务损失和释放的相关能量的数量）的观察要比损失更长的时间。基于这样的数据集，我们提出了希尔估计的改进版本。为了进行这种适应，目标变量和相关变量之间的尾部相关性起着重要的作用。我们建立了这个新估计量的渐近正态性。与Hill估计相比，它的行为得到了极大的改善，特别是渐近方差大大减少了，而我们可以使渐近偏差保持不变。仿真研究证实了我们改进后的估算器的性能大大提高。我们还提出了对上述地震损失的一种应用。