We investigate the asymptotic and finite sample behavior of the Hill estimator applied to time series contaminated by measurement or other errors. We show that for all discrete time models used in practice, whose non‐contaminated marginal distributions are regularly varying, the Hill estimator is consistent. Essentially, the only assumption on the errors is that they have lighter tails than the underlying unobservable process. The asymptotic justification however depends on the specific class of models assumed for the underlying unobservable process. We show by means of a simulation study that the asymptotic robustness of the Hill estimator is clearly manifested in finite samples. We further illustrate this robustness by a numerical study of the interarrival times of anomalies in a backbone internet network, the Internet2 in the United States; the anomalies arrival times are computed with a roundoff error.

中文翻译：

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用测量误差观察到的时间序列的 Hill 估计器的一致性

我们研究了应用于被测量或其他错误污染的时间序列的希尔估计器的渐近和有限样本行为。我们表明，对于实践中使用的所有离散时间模型，其未受污染的边际分布有规律地变化，希尔估计量是一致的。从本质上讲，对错误的唯一假设是它们的尾部比潜在的不可观察过程更轻。然而，渐近证明取决于为潜在的不可观察过程假设的特定模型类别。我们通过模拟研究表明，希尔估计器的渐近鲁棒性在有限样本中得到了明显的体现。我们通过对美国 Internet2 骨干互联网网络异常到达时间的数值研究进一步说明了这种稳健性；