Source: Ann. Statist.
Volume 36, Number 3
We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavy-tailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normal-score version outperforms traditional Gaussian likelihood ratio tests and their pseudo-Gaussian robustifications under a very broad range of non-Gaussian densities including, for instance, all multivariate Student and power-exponential distributions.
 Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
 Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. Roy. London Soc. Ser. A 160 268–282.
 Bartlett, M. S. and Kendall, D. G. (1946). The statistical analysis of variance-heterogeneity and the logarithmic transformation. Suppl. J. Roy. Statist. Soc. 8 128–138.
Mathematical Reviews (MathSciNet): MR19879
 Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647–671.
Mathematical Reviews (MathSciNet): MR663424
 Box, G. E. P. (1953). Non-normality and tests on variances. Biometrika 40 318–335.
Mathematical Reviews (MathSciNet): MR58937
 Cochran, W. G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Ann. Eugenics 11 47–52.
Mathematical Reviews (MathSciNet): MR5560
 Conover, W. J., Johnson, M. E. and Johnson, M. M. (1981). Comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23 351–361.
 Dümbgen, L. (1998). On Tyler’s M-functional of scatter in high dimension. Ann. Inst. Statist. Math. 50 471–491.
 Dümbgen, L. and Tyler, D. E. (2005). On the breakdown properties of some multivariate M-functionals. Scand. J. Statist. 32 247–264.
 Fligner, M. A. and Killeen, T. J. (1976). Distribution-free two-sample tests for scale. J. Amer. Statist. Assoc. 71 210–213.
Mathematical Reviews (MathSciNet): MR400532
 Goodnight, C. J. and Schwartz, J. M. (1997). A bootstrap comparison of genetic covariance matrices. Biometrics 53 1026–1039.
 Gupta, A. K. and Xu, J. (2006). On some tests of the covariance matrix under general conditions. Ann. Inst. Statist. Math. 58 101–114.
 Hájek, I. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325–346.
 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. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133.
 Hallin, M. and Paindaveine, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642–2678.
 Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707–2756.
 Hallin, M. and Paindaveine, D. (2006). Parametric and semiparametric inference for shape: The role of the scale functional. Statist. Decisions 24 1001–1023.
 Hallin, M. and Paindaveine, D. (2007). Optimal tests for homogeneity of covariance, scale, and shape. J. Multivariate Anal. To appear.
 Hallin, M. and Werker, B. J. M. (2003). Semiparametric efficiency, distribution-freeness, and invariance. Bernoulli 9 137–165.
 Hartley, H. O. (1950). The maximum F-ratio as a shortcut test for heterogeneity of variance. Biometrika 37 308–312.
 Heritier, S. and Ronchetti, E. (1994). Robust bounded-influence tests in general parametric models. J. Amer. Statist. Assoc. 89 897–904.
 Hettmansperger, T. P. and Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika 89 851–860.
 Jurečková, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40 1889–1900.
 Kreiss, J. P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112–133.
Mathematical Reviews (MathSciNet): MR885727
 Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
Mathematical Reviews (MathSciNet): MR856411
 Nagao, H. (1973). On some test criteria for covariance matrix. Ann. Statist. 1 700–709.
Mathematical Reviews (MathSciNet): MR339405
 Ollila, E., Hettmansperger, T. P. and Oja, H. (2004). Affine equivariant multivariate sign methods. Preprint, Univ. Jyväskylä.
 Paindaveine, D. (2006). A Chernoff–Savage result for shape. On the non-admissibility of pseudo-Gaussian methods. J. Multivariate Anal. 97 2206–2220.
 Paindaveine, D. (2007). A canonical definition of shape. Submitted.
 Perlman, M. D. (1980). Unbiasedness of the likelihood ratio tests for equality of several covariance matrices and equality of several multivariate normal populations. Ann. Statist. 8 247–263.
Mathematical Reviews (MathSciNet): MR560727
 Puri, M. L. and Sen, P. K. (1985). Nonparametric Methods in General Linear Models. Wiley, New York.
Mathematical Reviews (MathSciNet): MR794309
 Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263–1268.
 Salibian-Barrera, M., Van Aelst, S. and Willems, G. (2006). Principal components analysis based on multivariate MM-estimators with fast and robust bootstrap. J. Amer. Statist. Assoc. 101 1198–1211.
 Schott, J. R. (2001). Some tests for the equality of covariance matrices. J. Statist. Plann. Inference 94 25–36.
 Taskinen, S., Croux, C., Kankainen, A., Ollila, E. and Oja, H. (2006). Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices. J. Multivariate Anal. 97 359–384.
 Tatsuoka, K. S. and Tyler, D. E. (2000). On the uniqueness of S-functionals and M-functionals under nonelliptical distributions. Ann. Statist. 28 1219–1243.
 Tyler, D. E. (1983). Robustness and efficiency properties of scatter matrices. Biometrika 70 411–420.
Mathematical Reviews (MathSciNet): MR712028
 Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter. Ann. Statist. 15 234–251.
Mathematical Reviews (MathSciNet): MR885734
 Um, Y. and Randles, R. H. (1998). Nonparametric tests for the multivariate multi-sample location problem. Statist. Sinica 8 801–812.
 Yanagihara, H., Tonda, T. and Matsumoto, C. (2005). The effects of non-normality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption. J. Multivariate Anal. 96 237–264.
 Zhang, J. and Boos, D. D. (1992). Bootstrap critical values for testing homogeneity of covariance matrices. J. Amer. Statist. Assoc. 87 425–429.
 Zhu, L. X., Ng, K. W. and Jing, P. (2002). Resampling methods for homogeneity tests of covariance matrices. Statist. Sinica 12 769–783.