The weak limiting behavior of the de Haan-Resnick estimator of the exponent of a stable distribution

Abstract

The problem of estimating the exponent of a stable law received a considerable attention in the recent literature. Here, we deal with an estimate of such a exponent introduced by De Haan and Resnick when the corresponding distribution function belongs to the Gumbel's domain of attraction. This study permits to construct new statistical tests. Examples and simulations are given. The limiting law are shown to be the Gumbel's law and particular cases are given with norming constants expressed with iterated logarithms and exponentials.