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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 Mmax 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. [ABSTRACT FROM AUTHOR]