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Tests of independence among continuous random vectors based on Cramr-von Mises functionals of the empirical copula process

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  • Kojadinovic, Ivan
  • Holmes, Mark

Abstract

A decomposition of the independence empirical copula process into a finite number of asymptotically independent sub-processes was studied by Deheuvels. Starting from this decomposition, Genest and Rmillard recently investigated tests of independence among random variables based on Cramr-von Mises statistics derived from the sub-processes. A generalization of Deheuvels' decomposition to the case where independence is to be tested among continuous random vectors is presented. The asymptotic behavior of the resulting collection of Cramr-von Mises statistics is derived. It is shown that they are not distribution-free. One way of carrying out the resulting tests of independence then involves using the bootstrap or the permutation methodology. The former is shown to behave consistently, while the latter is employed in practice. Finally, simulations are used to study the finite-sample behavior of the tests.

Suggested Citation

  • Kojadinovic, Ivan & Holmes, Mark, 2009. "Tests of independence among continuous random vectors based on Cramr-von Mises functionals of the empirical copula process," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1137-1154, July.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1137-1154
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    References listed on IDEAS

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    1. Deheuvels, Paul, 1981. "An asymptotic decomposition for multivariate distribution-free tests of independence," Journal of Multivariate Analysis, Elsevier, vol. 11(1), pages 102-113, March.
    2. Ghoudi, Kilani & Kulperger, Reg J. & Rémillard, Bruno, 2001. "A Nonparametric Test of Serial Independence for Time Series and Residuals," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 191-218, November.
    3. Allaire, Jérôme & Lepage, Yves, 1991. "On a likelihood ratio test for independence," Statistics & Probability Letters, Elsevier, vol. 11(5), pages 449-452, May.
    4. Beran, R. & Bilodeau, M. & Lafaye de Micheaux, P., 2007. "Nonparametric tests of independence between random vectors," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1805-1824, October.
    5. Taskinen, Sara & Oja, Hannu & Randles, Ronald H., 2005. "Multivariate Nonparametric Tests of Independence," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 916-925, September.
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    Cited by:

    1. Mercadier, Cécile & Roustant, Olivier & Genest, Christian, 2022. "Linking the Hoeffding–Sobol and Möbius formulas through a decomposition of Kuo, Sloan, Wasilkowski, and Woźniakowski," Statistics & Probability Letters, Elsevier, vol. 185(C).
    2. Helmut Herwartz & Simone Maxand, 2020. "Nonparametric tests for independence: a review and comparative simulation study with an application to malnutrition data in India," Statistical Papers, Springer, vol. 61(5), pages 2175-2201, October.
    3. Lingyue Zhang & Dawei Lu & Xiaoguang Wang, 2020. "Measuring and testing interdependence among random vectors based on Spearman’s $$\rho $$ ρ and Kendall’s $$\tau $$ τ," Computational Statistics, Springer, vol. 35(4), pages 1685-1713, December.
    4. Berghaus, Betina & Segers, Johan, 2018. "Weak convergence of the weighted empirical beta copula process," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 266-281.
    5. González-Barrios, José M. & Gutiérrez-Peña, Eduardo & Rueda, Raúl, 2019. "A short note on the dependence structure of random vectors," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 200-205.
    6. repec:jss:jstsof:34:i09 is not listed on IDEAS
    7. Bianchi, Pascal & Elgui, Kevin & Portier, François, 2023. "Conditional independence testing via weighted partial copulas," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    8. Karel Hron & Paula Brito & Peter Filzmoser, 2017. "Exploratory data analysis for interval compositional data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(2), pages 223-241, June.
    9. Jin, Ze & Matteson, David S., 2018. "Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 304-322.
    10. Neumeyer, Natalie & Omelka, Marek & Hudecová, Šárka, 2019. "A copula approach for dependence modeling in multivariate nonparametric time series," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 139-162.
    11. Fan, Yanan & de Micheaux, Pierre Lafaye & Penev, Spiridon & Salopek, Donna, 2017. "Multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 189-210.
    12. Kojadinovic, Ivan, 2010. "Hierarchical clustering of continuous variables based on the empirical copula process and permutation linkages," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 90-108, January.
    13. Matieyendou Lamboni, 2024. "Kernel-based Measures of Association Between Inputs and Outputs Using ANOVA," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 790-826, August.
    14. Kalemkerian, Juan & Fernández, Diego, 2020. "An independence test based on recurrence rates," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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