We develop a new method for the statistical estimation of the tail of the distribution of earthquake sizes recorded in the Harvard catalog of seismic moments converted to m W -magnitudes (1977-2004 and 1977-2006). For this, we suggest a new parametric model for the distribution of main-shock magnitudes, which is composed of two branches, the pure Gutenberg-Richter distribution up to an upper magnitude threshold m 1 , followed by another branch with a maximum upper magnitude bound M max , which we refer to as the twobranch model. We find that the number of main events in the catalog (N = 3975 for 1977-2004 and N = 4193 for 1977-2006) is insufficient for a direct estimation of the parameters of this model, due to the inherent instability of the estimation problem. This problem is likely to be the same for any other two-branch model. This inherent limitation can be explained by the fact that only a small fraction of the empirical data populates the second branch. We then show that using the set of maximum magnitudes (the set of T-maxima) in windows of duration T days provides a significant improvement, in particular (i) by minimizing the negative impact of timeclustering of foreshock/main shock/aftershock sequences in the estimation of the tail of magnitude distribution, and (ii) by providing via a simulation method reliable estimates of the biases in the Moment estimation procedure (which turns out to be more efficient than the Maximum Likelihood estimation). We propose a method for the determination of the optimal choice of the T value minimizing the mean-squares-error of the estimation of the form parameter of the GEV distribution approximating the sample distribution of T-maxima, which yields T optimal = 500 days. We have estimated the following quantiles of the distribution of T-maxima for the whole period 1977-2006: Q 16% (M max ) = 9.3, Q 50% (M max ) = 9.7 and Q 84% (M max ) = 10.3. Finally, we suggest two more stable statistical characteristics of the tail of the distribution of earthquake magnitudes: The quantile Q T (q) of a high probability level q for the T-maxima, and the probability of exceedance of a high threshold magnitude q T (m*) = P{m k C m*}. We obtained the following sample estimates for the global Harvard catalogQ T ðq ¼ 0:98Þ ¼ 8:6AE0:2 andq T ð8Þ ¼ 0:13 À 0:20: The comparison between our estimates for the two periods 1977-2004 and 1977-2006, where the latter period included the great Sumatra earthquake 24.12.2004, m W = 9.0 confirms the instability of the estimation of the parameter M max and the stability of Q T (q) and q T (m*) = P{m k C m*}.

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The present work is a continuation and improvement of the method suggested in [Pisarenko et al. 2008] for the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. This distribution enables one to derive any desired statistical characteristic of the future T-maximum. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog and on the Fennoscandia catalog . For the global catalog, we obtain the following estimates: max M = 9.53 ± 0.52; ) 97 . 0 ( 10

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We have modified the new method for the statistical estimation of the tail distribution of earthquake seismic moments introduced by Pisarenko et al. (2009) and applied it to the earthquake catalog of Japan . The newly modified method is based on the two main limit theorems of the theory of extreme values and on the derived duality between the generalized Pareto distribution (GPD) and the generalized extreme value distribution (GEV). Using this method, we obtain the distribution of maximum earthquake magnitudes in future time intervals of arbitrary duration τ . This distribution can be characterized by its quantile Q q (τ ) at any desirable statistical level q. The quantile Q q (τ ) provides a much more stable and robust characteristic than the traditional absolute maximum magnitude M max (M max can be obtained as the limit of Q q (τ ) as q → 1, τ → ∞). The best estimates of the parameters governing the distribution of Q q (τ ) for Japan