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A multivariate version of Hoeffding's Phi-Square

  • Gaißer, Sandra
  • Ruppert, Martin
  • Schmid, Friedrich
Registered author(s):

    A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.

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

    Volume (Year): 101 (2010)
    Issue (Month): 10 (November)
    Pages: 2571-2586

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    Handle: RePEc:eee:jmvana:v:101:y:2010:i:10:p:2571-2586
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    1. Friedrich Schmid & Rafael Schmidt, 2007. "Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence," Metrika, Springer, vol. 66(3), pages 323-354, November.
    2. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: theory," Insurance: Mathematics and Economics, Elsevier, vol. 31(1), pages 3-33, August.
    3. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(01), pages 17-39, February.
    4. Harry Joe, 1989. "Estimation of entropy and other functionals of a multivariate density," Annals of the Institute of Statistical Mathematics, Springer, vol. 41(4), pages 683-697, December.
    5. Paul Doukhan & Jean-David Fermanian & Gabriel Lang, 2009. "An empirical central limit theorem with applications to copulas under weak dependence," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 65-87, February.
    6. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    7. Schmid, Friedrich & Schmidt, Rafael, 2007. "Multivariate extensions of Spearman's rho and related statistics," Statistics & Probability Letters, Elsevier, vol. 77(4), pages 407-416, February.
    8. Buhlmann, Peter & Kunsch, Hans R., 1999. "Block length selection in the bootstrap for time series," Computational Statistics & Data Analysis, Elsevier, vol. 31(3), pages 295-310, September.
    9. Schmid, Friedrich & Schmidt, Rafael, 2007. "Multivariate conditional versions of Spearman's rho and related measures of tail dependence," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1123-1140, July.
    10. Dhaene, J. & Denuit, M. & Goovaerts, M. J. & Kaas, R. & Vyncke, D., 2002. "The concept of comonotonicity in actuarial science and finance: applications," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 133-161, October.
    11. Jean-David FERMANIAN & Olivier SCAILLET, 2003. "Nonparametric Estimation of Copulas for Time Series," FAME Research Paper Series rp57, International Center for Financial Asset Management and Engineering.
    12. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
    13. M. Taylor, 2007. "Multivariate measures of concordance," Annals of the Institute of Statistical Mathematics, Springer, vol. 59(4), pages 789-806, December.
    14. Manuel Úbeda-Flores, 2005. "Multivariate versions of Blomqvist’s beta and Spearman’s footrule," Annals of the Institute of Statistical Mathematics, Springer, vol. 57(4), pages 781-788, December.
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