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We present a generic and powerful approach to study the statistics of extremephenomena (meteorology, finance, biology...) that we apply to the statisticalestimation of the tail of the distribution of earthquake sizes. The chiefinnovation is to combine the two main limit theorems of Extreme Value Theory(EVT) that allow us to derive the distribution of T-maxima (maximum magnitudeoccurring in sequential time intervals of duration T) for arbitrary T. Wepropose a method for the estimation of the unknown parameters involved in thetwo limit theorems corresponding to the Generalized Extreme Value distribution(GEV) and to the Generalized Pareto Distribution (GPD). We establish the directrelations between the parameters of these distributions, which permit toevaluate the distribution of the T-maxima for arbitrary T. The duality betweenthe GEV and GPD provides a new way to check the consistency of the estimationof the tail characteristics of the distribution of earthquake magnitudes forearthquake occurring over arbitrary time interval. We develop severalprocedures and check points to decrease the scatter of the estimates and toverify their consistency. We test our full procedure on the global Harvardcatalog (1977-2006) and on the Fennoscandia catalog (1900-2005). For the globalcatalog, 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 theGEV and GPD, including the most important form parameter, are also provided.

Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory