Generalized Pickands’ estimators for the tail index parameter and max-semistability

Abstract

Max-semistable laws arise as the limit of sequences of distribution functions of the form $F^{k_n}(a_nx+b_n)$, where {k _n } is a geometrically growing sequence. According to the results in Canto e Castro et al. (Theory Probab Appl 45:779–782, 2002), max-semistable models can be characterized by a parameter r (≥ 1), by the tail index parameter, and by a real function w defined on [0, log r]. A new problem is presented here motivated by the fact that the parameter r, which identifies each family of models, is not univocally determined. Namely, the same model is obtained using r or any of its positive integer powers. In this work, sequences of statistics that are ratios of certain differences of order statistics (like those in the Generalized Pickands’ estimator) will be considered. Their importance for the estimation of r will be emphasized along with the analysis of the asymptotic behaviour of their trajectories. Based on the obtained results, an heuristic method is proposed to estimate r which will permit, in a next step, the estimation of the tail index. Strengths and weaknesses of the methodology are made apparent through a simulation study.