Abstract—The paper addresses the problems associated with the maximum earthquakes in a seismically active region. Pisarenko and Rodkin (2009; 2010; 2015) proposed an alternative to the ambiguously determined parameter of the maximum regional magnitude M_max in the form of a clearly defined and statistically substantiated parameter of the maximum possible magnitude of an earthquake in a given region on a given future time interval T. In this work, we study the statistical characteristics of this parameter—quantiles with a given confidence level q. For the first time, the estimates for the bias, standard deviation, and root mean square deviation of these quantiles are conducted on a large set (1000) of independent samples (synthetic catalogs) with known distribution law. This allowed us to compare the true accuracy of the resulting estimates with that obtained in the case of using one catalog for which the distribution law is not known. The real efficiency of the quantile estimates is assessed from the examples with the exact known quantile values, and stability and robustness of these quantiles as an informative and important characteristic of seismic risk is demonstrated. The conducted comparisons make it possible to estimate the scatter of the quantiles in the region of large times T and with very high confidence levels q. The quantile estimates of the maximum possible events in a future time interval are substantially more robust at a stronger downward bend of the frequency–magnitude graph. A relationship is derived between the quantiles of a single earthquake and the quantiles of a maximum earthquake in the future time interval T. This relationship establishes the equivalence between the length of interval T and the level of significance (reliability) q, which in the case of heavy-tailed distributions should be taken into account in seismic risk assessment. The statistical analysis is carried out using two different methods of parameter estimation: by maximum likelihood estimation and by Bayesian approach. Both methods proved to be approximately equally efficient. Based on these methods, it is concluded that the estimate of the long-term seismic hazard for the regions with a distinctly pronounced downward bend of the frequency–magnitude graph is fairly stable and robust.

摘要—本文解决了与地震活跃地区的最大地震有关的问题。Pisarenko和Rodkin（2009; 2010; 2015）提出了一个替代方案，以明确定义并在统计上证实的最大区域震级M _max的给定区域内最大震级的参数的形式，对最大区域震级M _max的模棱两可确定的参数进行了替代。给定的未来时间间隔Ť。在这项工作中，我们研究了此参数的统计特性-具有给定置信度q的分位数。这些分位数的偏差，标准偏差和均方根偏差的估计首次在具有已知分布规律的大量独立样本（合成目录）上进行（1000个）。这使我们能够将得出的估计值的真实准确性与使用一个不知道其分布规律的目录所获得的估计值的真实准确性进行比较。从具有准确已知分位数的示例中评估分位数估计的实际效率，并证明这些分位数的稳定性和鲁棒性是地震风险的有益和重要特征。进行的比较使得有可能估计分位数在大时间T区域内的散布，并具有很高的置信度q。在频率幅度图的向下弯曲更强的情况下，未来时间间隔内最大可能事件的分位数估计会更加健壮。在将来的时间间隔T中，得出单个地震的分位数与最大地震的分位数之间的关系。这种关系建立了区间T的长度与有效水平（可靠性）q之间的等价关系，在重尾分布的情况下，应在地震风险评估中予以考虑。使用两种不同的参数估计方法进行统计分析：通过最大似然估计和通过贝叶斯方法。两种方法都被证明近似有效。基于这些方法，可以得出结论，对于频率-幅值图明显向下弯曲的区域，长期地震危险性的估计是相当稳定和稳健的。