  We present a generic and powerful approach to study the statistics of extreme
phenomena (meteorology, finance, biology...) that we apply to 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. 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 (1977-2006) and on the Fennoscandia catalog (1900-2005). For the global
catalog, we obtain the following estimates: Mmax = 9.53 +- 0.52;
quantile(0.97)==9.21 +- 0.20. For Fennoscandia, we obtain Mmax = 5.76 +- 0.165;
quantile(0.97) =5.44 +- 0.073. The estimates of all related parameters for the
GEV and GPD, including the most important form parameter, are also provided.
