Testing and estimation of the index of stability of univariate and bivariate symmetric $\alpha-$stable distributions via modified Greenwood statistic

TL;DR

This paper develops a new methodology using a modified Greenwood statistic for testing and estimating the stability index of univariate and bivariate symmetric alpha-stable distributions, outperforming classical methods.

Contribution

It extends the Greenwood statistic to bivariate symmetric alpha-stable distributions and introduces a novel testing approach for the stability index estimation.

Findings

Proposed methods outperform classical approaches in simulations.

Effective in distinguishing Gaussian from alpha-stable distributions near stability index 2.

Theoretical and practical data examples validate the methodology.

Abstract

We propose a testing and estimation methodology for univariate and bivariate symmatric $\alpha$-stable distributions using a modified version of the Greenwood statistic. Originally designed for positive-valued random variables, the Greenwood statistic, and its modified version tailored for symmetric distributions, have been predominantly applied to univariate random samples. In this paper, we extend the modified Greenwood statistic to a bivariate setting and examine its probabilistic properties within the class of $\alpha$-stable distributions, with a focus on the sub-Gaussian case. Additionally, we introduce a novel testing approach that considers two variations of the modified Greenwood statistic as test statistics for the bivariate case. In the univariate setting, we adapt the proposed testing methodology for estimating the stability index. The simulation studies presented demonstrate that our proposed methodology outperforms classical approaches previously used in this context and serves as an effective tool for distinguishing between Gaussian and $\alpha$-stable distributions with a stability index close to 2. The theoretical and simulation results are further illustrated with practical data examples.