Extremes occur in stationary regularly varying time series as short periods with several large observations, known as extremal blocks. We study cluster statistics summarizing the behavior of functions acting on these extremal blocks. Examples of cluster statistics are the extremal index, cluster size probabilities, and other cluster indices. The purpose of our work is twofold. First, we state the asymptotic normality of block estimators for cluster inference based on consecutive observations with large lp-norms, for p > 0. Second, we verify the conditions we require on classic models such as linear models and solutions of stochastic recurrence equations. Regarding linear models, we prove that the asymptotic variance of classical index cluster-based estimators is null as first conjectured in Hsing T. [26]. We illustrate our findings on simulations.
翻译:极端出现在固定时间序列中,以短期为固定时间序列,以若干大观测,称为极端区块。我们研究群集统计,总结在这些极端区块上发挥作用的功能行为。群集统计的例子有极端指数、群集大小概率和其他群集指数。我们的工作有两个目的。首先,我们指出基于与大低温连续观测的群集推断区块估计器的无症状正常性, p > 0。第二,我们核查我们要求的典型模型的条件,如线性模型和随机复发方程式的解决方案。关于线性模型,我们证明古典指数集集估计器的无症状差异与Hsing T. [26] 中的第一个预测是无效的。我们说明了我们在模拟方面的调查结果。