Annals of Statistics

Asymptotic local efficiency of Cramér–von Mises tests for multivariate independence

Christian Genest, Jean-François Quessy, and Bruno Rémillard

Full-text: Open access


Deheuvels [J. Multivariate Anal. 11 (1981) 102–113] and Genest and Rémillard [Test 13 (2004) 335–369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cramér–von Mises statistics derived from a Möbius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives.

Article information

Ann. Statist., Volume 35, Number 1 (2007), 166-191.

First available in Project Euclid: 6 June 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing 62G30: Order statistics; empirical distribution functions
Secondary: 62E20: Asymptotic distribution theory 60G15: Gaussian processes

Archimedean copula models asymptotic relative efficiency contiguous alternatives Cramér–von Mises statistics empirical copula process local power curve Möbius inversion formula tests of multivariate independence


Genest, Christian; Quessy, Jean-François; Rémillard, Bruno. Asymptotic local efficiency of Cramér–von Mises tests for multivariate independence. Ann. Statist. 35 (2007), no. 1, 166--191. doi:10.1214/009053606000000984.

Export citation


  • Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function. Ann. Math. Statist. 32 485–498.
  • Cotterill, D. S. and Csörgő, M. (1982). On the limiting distribution of and critical values for the multivariate Cramér–von Mises statistic. Ann. Statist. 10 233–244.
  • Cotterill, D. S. and Csörgő, M. (1985). On the limiting distribution of and critical values for the Hoeffding, Blum, Kiefer, Rosenblatt independence criterion. Statist. Decisions 3 1–48.
  • Csörgő, M. (1979). Strong approximations of the Hoeffding, Blum, Kiefer, Rosenblatt multivariate empirical process. J. Multivariate Anal. 9 84–100.
  • Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d'indépendance. Acad. Roy. Belg. Bull. Cl. Sci. (5) 65 274–292.
  • Deheuvels, P. (1981). An asymptotic decomposition for multivariate distribution-free tests of independence. J. Multivariate Anal. 11 102–113.
  • Deheuvels, P. and Martynov, G. V. (1996). Cramér–von Mises-type tests with applications to tests of independence for multivariate extreme-value distributions. Comm. Statist. Theory Methods 25 871–908.
  • Dugué, D. (1975). Sur des tests d'indépendance “indépendants de la loi”. C. R. Acad. Sci. Paris Sér. A-B 281 Aii, A1103–A1104.
  • Feuerverger, A. (1993). A consistent test for bivariate dependence. Internat. Statist. Rev. 61 419–433.
  • Gänßler, P. and Stute, W. (1987). Seminar on Empirical Processes. DMV Seminar 9. Birkhäuser, Basel.
  • Genest, C. and MacKay, R. J. (1986). Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 145–159.
  • Genest, C., Quessy, J.-F. and Rémillard, B. (2006). Local efficiency of a Cramér–von Mises test of independence. J. Multivariate Anal. 97 274–294.
  • Genest, C. and Rémillard, B. (2004). Tests of independence and randomness based on the empirical copula process. Test 13 335–369.
  • Ghoudi, K., Kulperger, R. J. and Rémillard, B. (2001). A nonparametric test of serial independence for time series and residuals. J. Multivariate Anal. 79 191–218.
  • Gieser, P. W. and Randles, R. H. (1997). A nonparametric test of independence between two vectors. J. Amer. Statist. Assoc. 92 561–567.
  • Gil-Pelaez, J. (1951). Note on the inversion theorem. Biometrika 38 481–482.
  • Hoeffding, W. (1948). A non-parametric test of independence. Ann. Math. Statist. 19 546–557.
  • Jing, P. and Zhu, L.-X. (1996). Some Blum–Kiefer–Rosenblatt type tests for the joint independence of variables. Comm. Statist. Theory Methods 25 2127–2139.
  • Kallenberg, W. C. M. and Ledwina, T. (1999). Data-driven rank tests for independence. J. Amer. Statist. Assoc. 94 285–301.
  • Nelsen, R. B. (1999). An Introduction to Copulas. Lecture Notes in Statist. 139. Springer, New York.
  • Shih, J. H. and Louis, T. A. (1996). Tests of independence for bivariate survival data. Biometrics 52 1440–1449.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.