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

TL;DR

This paper investigates the asymptotic behavior of the De Haan-Resnick estimator for the tail exponent of stable distributions within the Gumbel domain, enabling new statistical tests with theoretical and simulation support.

Contribution

It analyzes the weak limiting behavior of the De Haan-Resnick estimator for stable law exponents, providing new insights and tools for statistical testing.

Findings

Limiting distribution is Gumbel's law.

Norming constants involve iterated logarithms and exponentials.

Examples and simulations illustrate theoretical results.

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.