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

Kewin Pączek (), Damian Jelito (), Marcin Pitera () and Agnieszka Wyłomańska ()
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Kewin Pączek: Jagiellonian University
Damian Jelito: Jagiellonian University
Marcin Pitera: Jagiellonian University
Agnieszka Wyłomańska: Wrocław University of Science and Technology

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2024, vol. 33, issue 1, No 17, 297-334

Abstract: Abstract The class of $$\alpha $$ α -stable distributions is widely used in various applications, especially for modeling heavy-tailed data. Although the $$\alpha $$ α -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 distribution is close to Gaussian. In this paper, we propose a new estimation algorithm for the stability index, for samples from the symmetric $$\alpha $$ α -stable distribution. The proposed approach is based on a 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 ensemble methods. Although our focus is set on the symmetric $$\alpha $$ α -stable case, we demonstrate that the considered statistic is insensitive to the skewness parameter change, so our method could be also used in a more generic framework. For completeness, we also show how to apply our method to real data linked to financial market and plasma physics.

Keywords: Stable distribution; Heavy-tailed distribution; Conditional variance; Estimation; Tail index; Stability index; 62F10; 60E07; 62P35 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11749-023-00894-7

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