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On Kendall's Process


Author Info

  • Barbe, Philippe
  • Genest, Christian
  • Ghoudi, Kilani
  • Rémillard, Bruno


LetZ1, ..., Znbe a random sample of sizen[greater-or-equal, slanted]2 from ad-variate continuous distribution functionH, and letVi, nstand for the proportion of observationsZj,j[not equal to]i, such thatZj[less-than-or-equals, slant]Zicomponentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution functionKnderived from the (dependent) pseudo-observationsVi, n. This random quantity is a natural nonparametric estimator ofK, the distribution function of the random variableV=H(Z), whose expectation is an affine transformation of the population version of Kendall's tau in the cased=2. Since the sample version of[tau]is related in the same way to the mean ofKn, Genest and Rivest (1993,J. Amer. Statist. Assoc.) suggested that[formula]be referred to as Kendall's process. Weak regularity conditions onKandHare found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 58 (1996)
Issue (Month): 2 (August)
Pages: 197-229

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Handle: RePEc:eee:jmvana:v:58:y:1996:i:2:p:197-229

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Keywords: asymptotic calculations copulas dependent observations empirical processes Vapnik-Cervonenkis classes;


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Cited by:
  1. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, Elsevier, vol. 53(4), pages 391-399, July.
  2. Charpentier, Arthur & Segers, Johan, 2008. "Convergence of Archimedean copulas," Statistics & Probability Letters, Elsevier, Elsevier, vol. 78(4), pages 412-419, March.
  3. Christian Genest & Bruno Rémillard, 2004. "Test of independence and randomness based on the empirical copula process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, Springer, vol. 13(2), pages 335-369, December.
  4. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349,, revised Apr 2013.
  5. Ghoudi, Kilani & Kulperger, Reg J. & Rémillard, Bruno, 2001. "A Nonparametric Test of Serial Independence for Time Series and Residuals," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 79(2), pages 191-218, November.
  6. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
  7. Brechmann, Eike C. & Hendrich, Katharina & Czado, Claudia, 2013. "Conditional copula simulation for systemic risk stress testing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 722-732.
  8. Charpentier, A. & Segers, J.J.J., 2006. "Convergence of Archimedean Copulas," Discussion Paper, Tilburg University, Center for Economic Research 2006-28, Tilburg University, Center for Economic Research.
  9. Segers, Johan & Uyttendaele, Nathan, 2014. "Nonparametric estimation of the tree structure of a nested Archimedean copula," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 72(C), pages 190-204.
  10. Cousin, Areski & Di Bernardino, Elena, 2013. "On multivariate extensions of Value-at-Risk," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 119(C), pages 32-46.
  11. Jean-David Fermanian, 2012. "An overview of the goodness-of-fit test problem for copulas," Papers 1211.4416,
  12. Bedoui, Rihab & Ben Dbabis, Makram, 2009. "Copulas and bivariate Risk measures : an application to hedge funds," Economics Papers from University Paris Dauphine 123456789/3346, Paris Dauphine University.
  13. Elena Di Bernardino & Didier Rullière, 2014. "Estimation of multivariate critical layers: Applications to rainfall data," Working Papers hal-00940089, HAL.


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