Abstract

Heavy-tailed high-dimensional data are common in practice, including disease pre

diction, gene expression analysis, and risk management. Dependence measures for symmetric

α-stable (SαS) random vectors are commonly applied to heavy-tailed data. However, the existing

measures of dependence for symmetric α-stable (SαS) random vectors do not imply independence at zero. To address this problem, we introduce a novel measure of dependence, extended

codifference, for the SαS and non-symmetric α-stable heavy-tailed distribution family, that allows characterizing independence between heavy-tailed variables for 0 < α < 2. We propose an

efficient non-parametric estimator that does not require estimation of tail indices and obtain its

asymptotic distribution. Furthermore, we provide a guideline for selecting a suitable measure

of dependence based on the properties of each measure of association. Finally, we provide several simulation studies for further illustration and apply extended codifference to clustering of

single-cells based on their RNA-seq expression to identify cell types in adipose tissue.

Key words and phrases: Measure of dependence, stable random variables, infinite variance, codifference

Information

Preprint No.SS-2025-0370
Manuscript IDSS-2025-0370
Complete AuthorsVahed Maroufy, Mohsen Rezapour, Mahmoud Zarepour, Bahareh Afhami
Corresponding AuthorsVahed Maroufy
Emailsvahed.maroufy@uth.tmc.edu

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Acknowledgments

Dr. Maroufy and Rezapour were in part supported by Dr. Maroufy’s faculty start-up

funds from The University of Texas Health Science Center at Houston. Dr. Zarepur

was supported by NSERC grant RGPIN/ 2018-04008.

Supplementary Materials

The online Supplementary Material contains selected graphs for the simulation example,

along with proofs and detailed derivations of the theorems and remarks.

Supplementary materials are available for download.