Tail Inference for a Law in a Max-Semistable Domain of Attraction

Abstract

The class of max-semistable distributions appeared in the literature of extremes, in a work of Pancheva (1992), as the limit distribution of samples with size growing geometrically with ratio r > 1. In Canto e Castro et al. (2002) it is proved that any max-semistable distribution function has a logperiodic component and can be characterized by the period therein, by a tail index parameter and by a real function y representing a repetitive pattern. Statistical inference in the max-semistable setup can be performed through convenient sequences of generalized Pickands' statistics, depending on a tuning parameter s. More precisely, in order to obtain estimators for the period and for the tail index, we can use the fact that the mentioned sequences converge in probability only when s = r (or any of its integer powers), having an oscillatory behavior otherwise. This work presents a procedure to estimate the function y as well as high quantiles. The suggested methodologies are applied to real data consisting in seismic moments of major earthquakes in the Pacific Region.