Rare events in a log-Weibull scenario - Application to earthquake magnitude data

We discuss the pertinency of the log-Weibull model in the statistical understanding of energy release for earthquake magnitude data. This model has many interesting features, the most remarkable of which being: depending on the value of the deformation index of the source, it may present tails ranging from moderately heavy () to very heavy (with tail index zero as ), through hyperbolic (power law) for the critical value . Under this model (for which a precise tail study is supplied), the occurrence of power laws appears as a critical phenomenon: this reinforces the current trend predicting that some departure from the ideal (strictly scaling fractal) model may be ubiquitous. Having applied an affine transformation in the logarithmic scale, quantile estimation and the Kolmogorov-Smirnov statistics are used to fit the log-Weibull distribution to a realization of an iid sample. This enables to decide whether the upper tail of the phenomenon under study is light/heavy/very heavy. A comparative study of recorded French and Japanese earthquake magnitudes suggests that they exhibit comparable tail behaviour, albeit with different centrality and dispersion parameters. Received 15 March 1999 and Received in final form 20 May 1999