Abstract
Several tests for multivariate mean vector have been proposed in the recent literature. Generally, these tests are directly concerned with the mean vector of a high-dimensional distribution. The paper presents two new test procedures for testing mean vector in large dimension and small samples. We do not focus on the mean vector directly, which is a different framework from the existing choices. The first test procedure is based on the asymptotic distribution of the test statistic, where the dimension increases with the sample size. The second test procedure is based on the permutation distribution of the test statistic, where the sample size is fixed and the dimension grows to infinity. Simulations are carried out to examine the finite-sample performance of the tests and to compare them with some popular nonparametric tests available in the literature.
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References
Andrews, D. Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory, 4: 458–467 (1988).
Bai, Z., Sarandasa, H. Effect of high dimension: by an example of a two sample problem. Statistica Sinica, 6: 311–329 (1996)
Biswas, M., Mukhopadhyay, M., Ghosh, A. A distribution-free two-sample run test applicable to high-dimensional data. Biometrika, 101: 913–926 (2014)
Buja, A., Logan, B., Reeds, J. Inequalities and positive definite functions arising from a problem in multi dimensional scaling. Annals of Statistics, 22: 406–438 (1994)
Cai, T., Liu, W. and **a, Y. Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society, Series B, 76: 349–372 (2014)
Chang, J., Zheng, C., Zhou, W. X., Zhou, W. Simulation-based hypothesis testing of high dimensional means under covariance heterogeneity. Biometrics, 73: 1300–1310 (2017)
Chen, S., Qin, Y. A two sample test for high dimensional data with applications to gene-set testing. Annals of Statistics, 38: 808–835 (2010)
Chen, S., Li, J., Zhong, P. Two-Sample and ANOVA Tests for High Dimensional Means. Annals of Statistics, 47: 1443–1474
Hall, P., Heyde, C. Martingale Limit Theory and Its Application. Academic Press, New York, 1980
Hall, P., Marron, J., Neeman, A. Geometric representation of high dimension, low sample size data. Journal of the Royal Statistical Society, Series B, 67: 427–444 (2005)
Ilmonen, P., Paindaveine, D. Semiparametrically efficient inference based on signed ranks in symmetric independent component models. Annals of Statistics, 39: 2448–2476 (2011)
Park, J., Nag Ayyala, D. A test for the mean vector in large dimension and small samples. Journal of Statistical Planning and Inference, 143: 929–943 (2013)
Székely, G., Rizzo, M. Energy statistics: A class of statistics based on distances. Journal of Statistical Planning and Inference, 143: 1249–1272
Székely, G., Rizzo, M. The distance correlation t-test of independence in high dimension. Journal of Multivariate Analysis, 117: 193–213 (2013)
Srivastava, M. A test for the mean vector with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 100: 518–532 (2009)
Srivastava, M., Du, M. A test for the mean vector with fewer observations than the dimension. Journal of Multivariate Analysis, 99: 386–402
Xu, K., Hao, X. A nonparametric test for block-diagonal covariance structure in high dimension and small samples. Journal of Multivariate Analysis, 173: 551–567 (2019)
Xue, K., Yao, F. Distribution and correlation-free two-sample test of high-dimensional means. The Annals of Statistics, 48: 1304–1328
Yao, S., Zhang, X., Shao, X. Testing mutual independence in high dimension via distance covariance. Journal of Royal Statistical Society, Series B, 80: 455–480 (2018)
Zhong, P., Chen, S. Tests for high dimensional regression coefficients with factorial designs. Journal of the American Statistical Association, 106: 260–274 (2011)
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The authors thank the editor, the AE, and the reviewers for their constructive comments, which have led to a dramatic improvement of the earlier version of this article.
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Chen, B., Wang, Hm. High-dimensional Tests for Mean Vector: Approaches without Estimating the Mean Vector Directly. Acta Math. Appl. Sin. Engl. Ser. 38, 78–86 (2022). https://doi.org/10.1007/s10255-022-1070-z
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DOI: https://doi.org/10.1007/s10255-022-1070-z