Institute of Mathematical Statistics Collections
Chernoff-Savage and Hodges-Lehmann results for Wilks’ test of multivariate independence
We extend to rank-based tests of multivariate independence the Chernoff-Savage and Hodges-Lehmann classical univariate results. More precisely, we show that the Taskinen, Kankainen and Oja (2004) normal-score rank test for multivariate independence uniformly dominates – in the Pitman sense – the classical Wilks (1935) test, which establishes the Pitman non-admissibility of the latter, and provide, for any fixed space dimensions p, q of the marginals, the lower bound for the asymptotic relative efficiency, still with respect to Wilks’ test, of the Wilcoxon version of the same.
First available in Project Euclid: 1 April 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Copyright © 2008, Institute of Mathematical Statistics
Hallin, Marc; Paindaveine, Davy. Chernoff-Savage and Hodges-Lehmann results for Wilks’ test of multivariate independence. Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, 184--196, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/193940307000000130. https://projecteuclid.org/euclid.imsc/1207058273
-  Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests. Ann. Math. Statist. 29 972–994.
-  Gastwirth, J. L. and Wolff, S. S. (1968). An elementary method for obtaining lower bounds on the asymptotic power of rank tests. Ann. Math. Statist. 39 2128–2130.
-  Gieser, P. W. and Randles, R. H. (1997). A nonparametric test of independence between two vectors. J. Amer. Statist. Assoc. 92 561–567.
-  Hájek, I., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, New York.
-  Hallin, M. (1994). On the Pitman-nonadmissibility of correlogram-based methods. J. Time Ser. Anal. 15 607–612.
-  Hallin, M., Oja, H. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape: II. Optimal R-estimation of shape. Ann. Statist. 34 2757–2789.
-  Hallin, M. and Paindaveine, D. (2002a). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133.
-  Hallin, M., and Paindaveine, D. (2002b). Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence. Bernoulli 8 787–816.
-  Hallin, M. and Paindaveine, D. (2002c). Multivariate signed ranks: Randles’ interdirections or Tyler’s angles? In Statistical Data Analysis Based on the L1 Norm and Related Procedures (Y. Dodge, ed.) 271–282. Birkhäuser, Basel.
-  Hallin, M. and Paindaveine, D. (2005). Affine invariant aligned rank tests for the multivariate general linear model with ARMA errors. J. Multivariate Anal. 93 122–163.
-  Hallin, M. and Paindaveine, D. (2006a). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707–2756.
-  Hallin, M. and Paindaveine, D. (2006b). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 1001–1023.
-  Hallin, M. and Puri, M. L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal. 50 175–237.
-  Hallin, M. and Tribel, O. (2000). The efficiency of some nonparametric rank-based competitors to correlogram methods. In Game Theory, Optimal Stopping, Probability, and Statistics (F. T. Bruss and L. Le Cam, eds.). Papers in Honor of T.S. Ferguson on the Occasion of his 70th Birthday. I.M.S. Lecture Notes-Monograph Series 249–262.
-  Hodges, J. L. and Lehmann, E. L. (1956). The efficiency of some nonparametric competitors of the t-test. Ann. Math. Statist. 27 324–335.
-  Hotelling, H. (1931). The generalization of Student’s ratio. Ann. Math. Statist. 2 360–378.
-  Hotelling, H. and Pabst, M. R. (1936). Rank correlation and tests of significance involving no assumption of normality. Ann. Math. Statist. 7 29–43.
-  Kendall, M. G. (1938). A new measure of rank correlation. Biometrika 30 81–93.
-  Konijn, H. S. (1956). On the power of certain tests for independence in bivariate populations. Ann. Math. Statist. 27 300–323.
-  Muirhead, R. J. and Waternaux, C. M. (1980). Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika 67 31–43.
-  Oja, H. and Paindaveine, D. (2005). Optimal signed-rank tests based on hyperplanes. J. Statist. Plann. Inference 135 300–323.
-  Paindaveine, D. (2004). A unified and elementary proof of serial and nonserial, univariate and multivariate, Chernoff-Savage results. Statist. Methodology 1 81–91.
-  Paindaveine, D. (2006). A Chernoff-Savage result for shape. On the nonadmissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206–2220.
-  Paindaveine, D. (2008). A canonical definition of shape. Statist. Probab. Lett. To appear.
-  Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. Wiley, New York.
-  Puri, M. L. and Sen, P. K. (1985). Nonparametric Methods in General Linear Models. Wiley, New York.
-  Randles, R. H. (1984). On tests applied to residuals. J. Amer. Statist. Assoc. 79 349–354.
-  Randles, R. H. (1989). A distribution-free multivariate test based on interdirections. J. Amer. Statist. Assoc. 84 1045–1050.
-  Spearman, C. (1904). The proof and measurement of association between two things. Amer. J. Psychology 15 72–101.
-  Taskinen, S., Kankainen, A. and Oja, H. (2003). Sign test of independence between two random vectors. Statist. Probab. Lett. 62 9–21.
-  Taskinen, S., Kankainen, A. and Oja, H. (2004). Rank scores tests of multivariate independence. In Theory and Applications of Recent Robust Methods (M. Hubert, G. Pison, A. Struyf and S. Van Aelst, eds.) 329–342. Birkhäuser, Basel.
-  Taskinen, S., Oja, H. and Randles, R. (2005). Multivariate nonparametric tests of independence. J. Amer. Statist. Assoc. 100 916–925.
-  Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin 1 80–83.
-  Wilks, S. S. (1935). On the independence of k sets of normally distributed statistical variables. Econometrica 3 309–326.