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We develop a new method for the statistical esitmation of the tail of thedistribution of earthquake sizes recorded in the Worldwide Harvard catalog ofseismic moments converted to mW-magnitudes (1977-2004 and 1977-2006). We showthat using the set of maximum magnitudes (the set of T-maxima) in windows ofduration T days provides a significant improvement over existing methods, inparticular (i) by minimizing the negative impact of time-clustering offoreshock / main shock /aftershock sequences in the estimation of the tail ofthe magnitude distribution, and (ii) by providing via a simulation methodreliable estimates of the biases in the Moment estimation procedure (whichturns out to be more efficient than the Maximum Likelihood estimation). Using asimulation method, we have determined the optimal window size of the T-maximato be T=500 days. We have estimated the following quantiles of the distributionof T-maxima of earthquake magnitudes for the whole period 1977-2006:Q_{0.16}(Mmax)=9.3, Q_{0.5}(Mmax)=9.7 and Q_{0.84}(Mmax)=10.3. Finally, wesuggest two more stable statistical characterristics of the tail of thedistribution of earthquake magnitudes: the quantile QT(q) of a high probabilitylevel q for the T-maxima, and the probability of exceedence for a highthreshold magnitude. We obtained the following sample estimates for the globalHarvard catalog: QT(q=0.98)=8.6 +- 0.2 and a probability for the T-maxima toexceed magnitude 8 equal to 0.13-0.20. The comparison between our estimates forthe two periods 1977-2004 and 1977-2006, where the later period includes thegreat Sumatra earthquake, 24.12.2004, mW=9.0, confirms the instability of theestimation of the parameter Mmax and the stability of the two other estimates.Comment: 40 pages including 16 figures and 7 table

New Approach to the Characterization of M max and of the Tail of the Distribution of Earthquake Magnitudes