Abstract
Schmid and Schmidt (Stat Probab Lett 77:407–416, 2007) recently described multivariate extensions to the population and sample versions of Spearman’s rank correlation coefficient. In this paper, the asymptotic relative efficiency of tests for multivariate independence based on the latter measures are explicitly computed under many distributional scenarios based on copula models, which complement findings made by Stepanova (Math Methods Stat 12:197–217, 2003). New Spearman-type statistics based on the so-called Möbius decomposition of the empirical copula process and of the survival copula process are also investigated in the light of their asymptotic relative efficiencies.
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References
Bakirov NK, Rizzo ML, Székely GJ (2006) A multivariate nonparametric test of independence. J Multivar Anal 97: 1742–1756
Binet A, Vaschide N (1897) Corrélation des épreuves physiques. L’année Psychologique 4: 142–172
Blum JR, Kiefer J, Rosenblatt M (1961) Distribution free tests of independence based on the sample distribution function. Ann Math Stat 32: 485–498
Deheuvels P (1979) La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance. Acad Royal Bel, Bull Class Sci 65: 274–292
Deheuvels P (1981) An asymptotic decomposition for multivariate distribution-free tests of independence. J Multivar Anal 11: 102–113
Dugué D (1975) Sur des tests d’indépendance “indépendants de la loi”. C R Acad Sci Paris Sér A-B 281: A1103–A1104
Fermanian J-D, Radulović D, Wegkamp M (2004) Weak convergence of empirical copula processes. Bernoulli 10: 847–860
Gaenssler P., Stute W. (1987) Seminar on empirical processes. DMV Seminar, vol 9. Birkhäuser Verlag, Basel
Genest C, Rémillard B (2004) Tests of independence and randomness based on the empirical copula process. Test 13: 335–370
Genest C, Quessy J-F, Rémillard B (2006) Local efficiency of a Cramér–von Mises test of independence. J Multivar Anal 97: 274–294
Genest C, Quessy J-F, Rémillard B (2007) Asymptotic local efficiency of Cramér–von Mises tests for multivariate independence. Ann Stat 35: 166–191
Ghoudi K, Kulperger RJ, Rémillard B (2001) A nonparametric test of serial independence for time series and residuals. J Multivar Anal 79: 191–218
Gieser PW, Randles RH (1997) A nonparametric test of independence between two vectors. J Amer Stat Assoc 92: 561–567
Hájek J, Šidák Z (1967) Theory of rank tests. Academic Press, New York
Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19: 293–325
Joe H. (1997) Multivariate models and dependence concepts. Monographs on statistics and applied probability, vol 73. Chapman & Hall, London
Joe H (1990) Multivariate concordance. J Multivar Anal 35: 12–30
Lee AJ. (1990) U-statistics. Theory and practice. Statistics: textbooks and monographs, vol 110. Marcel Dekker, Inc., New York
Nelsen RB. (1999) An introduction to copulas. Lecture notes in statistics, vol 139. Springer, New York
Nikitin YY, Pankrashova AG (1990) Bahadur efficiency and local asymptotic optimality of some nonparametric independence tests. J Soviet Math 52: 2942–2955
Ruymgaart FH, Zuijlen MCA (1978) Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case. Ann Stat 6: 588–602
Schmid F, Schmidt R (2007) Multivariate extensions of Spearman’s rho and related statistics. Stat Probab Lett 77: 407–416
Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15: 88
Stepanova NA (2003) Multivariate rank tests for independence and their asymptotic efficiency. Math Methods Stat 12: 197–217
Taskinen S, Oja H, Randles RH (2005) Multivariate nonparametric tests of independence. J Am Stat Assoc 100: 916–925
Tsukahara H (2005) Semiparametric estimation in copula models. Can J Stat 33: 357–375
van der Vaart AW., Wellner JA. (1996) Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer, New York
Wolff EF (1980) n-Dimensional measures of dependence. Stochastica 4: 175–188
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Quessy, JF. Theoretical efficiency comparisons of independence tests based on multivariate versions of Spearman’s rho. Metrika 70, 315–338 (2009). https://doi.org/10.1007/s00184-008-0194-3
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DOI: https://doi.org/10.1007/s00184-008-0194-3