Kendall's tau and Spearman's rho for n-dimensional Archimedean copulas and their asymptotic properties

We derive formulas for the dependence measures and for Archimedean n-copulas. These measures are n-dimensional analogues of the popular nonparametric dependence measures: Kendall's tau and Spearman's rho. For we obtain two formulas, both involving integrals of univariate functions. The for...

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Bibliographic Details
Published in:Journal of nonparametric statistics 2015-10, Vol.27 (4), p.442-459
Main Author: Wysocki, Włodzimierz
Format: Article
Language:English
Subjects:
Additives
Archimedean n-copula
Asymptotic methods
Asymptotic properties
Cubes
Generators
infinite-dimensional Archimedean n-copula
Integrals
Kendall's tau
Laplace transform
Laplace transforms
Nonparametric statistics
Primary: 62H15
Secondary: 62G20
Spearman's rho
Williamson n-transform
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Summary:We derive formulas for the dependence measures and for Archimedean n-copulas. These measures are n-dimensional analogues of the popular nonparametric dependence measures: Kendall's tau and Spearman's rho. For we obtain two formulas, both involving integrals of univariate functions. The formulas for involve integrals of n-variate functions. We also obtain formulas for the three measures for copulas whose additive generators have completely monotone inverses. These formulas feature integrals of 2-variate functions (we use the Laplace transform). We study the asymptotic properties of the sequences and , for a sequence of Archimedean copulas with a common additive generator. We also investigate the limit of this sequence, which is an infinite-dimensional copula on the Hilbert cube.
ISSN:1048-5252
1029-0311
DOI:10.1080/10485252.2015.1070849