Estimation of stability index for symmetric α-stable distribution using quantile conditional variance ratios

Abstract: The class of $\alpha$-stable distributions is widely used in various applications, especially for modelling heavy-tailed data. Although the $\alpha$-stable distributions have been used in practice for many years, new methods for identification, testing, and estimation are still being refined and new approaches are being proposed. The constant development of new statistical methods is related to the low efficiency of existing algorithms, especially when the underlying sample is small or the underlying distribution is close to Gaussian. In this paper we propose a new estimation algorithm for stability index, for samples from the symmetric $\alpha$-stable distribution. The proposed approach is based on quantile conditional variance ratio. We study the statistical properties of the proposed estimation procedure and show empirically that our methodology often outperforms other commonly used estimation algorithms. Moreover, we show that our statistic extracts unique sample characteristics that can be combined with other methods to refine existing methodologies via ensamble methods. Although our focus is set on the symmetric $\alpha$-stable case, we demonstrate that the considered statistic is insensitive to the skewness parameter change, so that our method could be also used in a more generic framework. For completeness, we also show how to apply our method on real data linked to plasma physics.