  We develop a new method for the statistical esitmation of the tail of the
distribution of earthquake sizes recorded in the Worldwide Harvard catalog of
seismic moments converted to mW-magnitudes (1977-2004 and 1977-2006). We show
that using the set of maximum magnitudes (the set of T-maxima) in windows of
duration T days provides a significant improvement over existing methods, in
particular (i) by minimizing the negative impact of time-clustering of
foreshock / main shock /aftershock sequences in the estimation of the tail of
the 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). Using a
simulation method, we have determined the optimal window size of the T-maxima
to be T=500 days. We have estimated the following quantiles of the distribution
of 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, we
suggest two more stable statistical characterristics of the tail of the
distribution of earthquake magnitudes: the quantile QT(q) of a high probability
level q for the T-maxima, and the probability of exceedence for a high
threshold magnitude. We obtained the following sample estimates for the global
Harvard catalog: QT(q=0.98)=8.6 +- 0.2 and a probability for the T-maxima to
exceed magnitude 8 equal to 0.13-0.20. The comparison between our estimates for
the two periods 1977-2004 and 1977-2006, where the later period includes the
great Sumatra earthquake, 24.12.2004, mW=9.0, confirms the instability of the
estimation of the parameter Mmax and the stability of the two other estimates.
