Tests based on mean vectors and spatial signs and ranks for a zero mean in one-sample problems and for the equality of means in two-sample problems have been studied in the recent literature for high-dimensional data with the dimension larger than the sample size. For the above testing problems, we show that under suitable sequences of alternatives, the powers of the mean-based tests and the tests based on spatial signs and ranks tend to be same as the data dimension tends to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setting. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial-sign- and rank-based tests for high-dimensional data can be asymptotically more powerful than the mean-based tests if, in addition to the data dimension, the sample size also tends to infinity. The sizes of some mean-based tests for high-dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.
Ann. Statist.
45(2):
771-799
(April 2017).
DOI: 10.1214/16-AOS1467
Cai, T. T., Liu, W. and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 349–372.Cai, T. T., Liu, W. and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 349–372.
Chakraborty, A. and Chaudhuri, P. (2016). Supplement to “Tests for high-dimensional data based on means, spatial signs and spatial ranks.” DOI:10.1214/16-AOS1467SUPP.Chakraborty, A. and Chaudhuri, P. (2016). Supplement to “Tests for high-dimensional data based on means, spatial signs and spatial ranks.” DOI:10.1214/16-AOS1467SUPP.
Gregory, K. B., Carroll, R. J., Baladandayuthapani, V. and Lahiri, S. N. (2015). A two-sample test for equality of means in high dimension. J. Amer. Statist. Assoc. 110 837–849.Gregory, K. B., Carroll, R. J., Baladandayuthapani, V. and Lahiri, S. N. (2015). A two-sample test for equality of means in high dimension. J. Amer. Statist. Assoc. 110 837–849.
Hettmansperger, T. P. and McKean, J. W. (2011). Robust Nonparametric Statistical Methods, 2nd ed. Monographs on Statistics and Applied Probability 119. CRC Press, Boca Raton, FL.Hettmansperger, T. P. and McKean, J. W. (2011). Robust Nonparametric Statistical Methods, 2nd ed. Monographs on Statistics and Applied Probability 119. CRC Press, Boca Raton, FL.
Katayama, S. and Kano, Y. (2014). A new test on high-dimensional mean vector without any assumption on population covariance matrix. Comm. Statist. Theory Methods 43 5290–5304.Katayama, S. and Kano, Y. (2014). A new test on high-dimensional mean vector without any assumption on population covariance matrix. Comm. Statist. Theory Methods 43 5290–5304.
Kolmogorov, A. N. and Rozanov, J. A. (1960). On a strong mixing condition for stationary Gaussian processes. Teor. Verojatnost. i Primenen. 5 222–227.Kolmogorov, A. N. and Rozanov, J. A. (1960). On a strong mixing condition for stationary Gaussian processes. Teor. Verojatnost. i Primenen. 5 222–227.
Marden, J. I. (1999). Multivariate rank tests. In Multivariate Analysis, Design of Experiments, and Survey Sampling. Statist. Textbooks Monogr. 159 401–432. Dekker, New York.Marden, J. I. (1999). Multivariate rank tests. In Multivariate Analysis, Design of Experiments, and Survey Sampling. Statist. Textbooks Monogr. 159 401–432. Dekker, New York.
Oja, H. (2010). Multivariate Nonparametric Methods with R: An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics 199. Springer, New York.Oja, H. (2010). Multivariate Nonparametric Methods with R: An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics 199. Springer, New York.
Wei, S., Lee, C., Wichers, L. and Marron, J. S. (2016). Direction-projection-permutation for high-dimensional hypothesis tests. J. Comput. Graph. Statist. 25 549–569.Wei, S., Lee, C., Wichers, L. and Marron, J. S. (2016). Direction-projection-permutation for high-dimensional hypothesis tests. J. Comput. Graph. Statist. 25 549–569.
