Linear hypothesis testing in linear models with high-dimensional responses. (English) Zbl 1515.62057
Summary: In this article, we propose a new projection test for linear hypotheses on regression coefficient matrices in linear models with high-dimensional responses. We systematically study the theoretical properties of the proposed test. We first derive the optimal projection matrix for any given projection dimension to achieve the best power and provide an upper bound for the optimal dimension of projection matrix. We further provide insights into how to construct the optimal projection matrix. One- and two-sample mean problems can be formulated as special cases of linear hypotheses studied in this article. We both theoretically and empirically demonstrate that the proposed test can outperform the existing ones for one- and two-sample mean problems. We conduct Monte Carlo simulation to examine the finite sample performance and illustrate the proposed test by a real data example.
MSC:
| 62H15 | Hypothesis testing in multivariate analysis |
| 62J15 | Paired and multiple comparisons; multiple testing |
References:
| [1] | Aoshima, M.; Yata, K., “Two-Sample Tests for High-Dimension, Strongly Spiked Eigenvalue Models, Statistica Sinica, 28, 43-62 (2018) · Zbl 1382.62028 · doi:10.5705/ss.202016.0063 |
| [2] | Ashburner, M.; Ball, C. A.; Blake, J. A.; Botstein, D.; Butler, H.; Cherry, J. M.; Davis, A. P.; Dolinski, K.; Dwight, S. S.; Eppig, J. T.; Harris, M. A., “Gene Ontology: Tool for the Unification of Biology, Nature Genetics, 25, 25-29 (2000) · doi:10.1038/75556 |
| [3] | Bai, Z.; Saranadasa, H., “Effect of High Dimension: By an Example of a Two Sample Problem, Statistica Sinica, 6, 311-329 (1996) · Zbl 0848.62030 |
| [4] | Baik, J.; Silverstein, J. W., “Eigenvalues of Large Sample Covariance Matrices of Spiked Population Models, Journal of Multivariate Analysis, 97, 1382-1408 (2006) · Zbl 1220.15011 · doi:10.1016/j.jmva.2005.08.003 |
| [5] | Benjamini, Y.; Hochberg, Y., “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing, Journal of the Royal Statistical Society, Series B, 57, 289-300 (1995) · Zbl 0809.62014 · doi:10.1111/j.2517-6161.1995.tb02031.x |
| [6] | Benjamini, Y.; Yekutieli, D., “The Control of the False Discovery Rate in Multiple Testing Under Dependency, The Annals of Statistics, 29, 1165-1188 (2001) · Zbl 1041.62061 · doi:10.1214/aos/1013699998 |
| [7] | Cai, T.; Liu, W.; Xia, Y., “Two-Sample Test of High Dimensional Means Under Dependence, Journal of the Royal Statistical Society, Series B, 76, 349-372 (2014) · Zbl 07555454 · doi:10.1111/rssb.12034 |
| [8] | Chen, S. X.; Li, J.; Zhong, P.-S., “Two-Sample and ANOVA Tests for High Dimensional Means, The Annals of Statistics, 47, 1443-1474 (2019) · Zbl 1417.62147 · doi:10.1214/18-AOS1720 |
| [9] | Chen, S. X.; Qin, Y.-L., “A Two-Sample Test for High-Dimensional Data With Applications to Gene-Set Testing, The Annals of Statistics, 38, 808-835 (2010) · Zbl 1183.62095 · doi:10.1214/09-AOS716 |
| [10] | Donoho, D. L.; Gavish, M.; Johnstone, I. M., “Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model, The Annals of Statistics, 46, 1742 (2018) · Zbl 1403.62099 · doi:10.1214/17-AOS1601 |
| [11] | Fan, J.; Liao, Y.; Yao, J., “Power Enhancement in High-Dimensional Cross-Sectional Tests, Econometrica, 83, 1497-1541 (2015) · Zbl 1410.62201 · doi:10.3982/ECTA12749 |
| [12] | He, Y.; Jiang, T.; Wen, J.; Xu, G., “Likelihood Ratio Test in Multivariate Linear Regression: From Low to High Dimension,”, Statistica Sinica (2019) |
| [13] | He, Y.; Xu, G.; Wu, C.; Pan, W., “Asymptotically Independent U-Statistics in High-Dimensional Testing, The Annals of Statistics, 49, 154-181 (2021) · Zbl 1461.62233 · doi:10.1214/20-AOS1951 |
| [14] | Hotelling, H., “The Generalization of Student’s Ratio, The Annals of Mathematical Statistics, 2, 360-378 (1931) · JFM 57.0633.01 · doi:10.1214/aoms/1177732979 |
| [15] | Huang, Y. (2015) |
| [16] | Johnstone, I. M., “On the Distribution of the Largest Eigenvalue in Principal Components Analysis, The Annals of Statistics, 29, 295-327 (2001) · Zbl 1016.62078 · doi:10.1214/aos/1009210544 |
| [17] | Kock, A. B.; Preinerstorfer, D., “Power in High-Dimensional Testing Problems, Econometrica, 87, 1055-1069 (2019) · Zbl 1420.62253 · doi:10.3982/ECTA15844 |
| [18] | Lopes, M.; Jacob, L.; Wainwright, M. J., “A More Powerful Two-Sample Test in High Dimensions Using Random Projection, Advances in Neural Information Processing Systems, 1206-1214 (2011) |
| [19] | Srivastava, M. S.; Du, M., “A Test for the Mean Vector With Fewer Observations Than the Dimension, Journal of Multivariate Analysis, 99, 386-402 (2008) · Zbl 1148.62042 · doi:10.1016/j.jmva.2006.11.002 |
| [20] | Srivastava, R.; Li, P.; Ruppert, D., “RAPTT: An Exact Two-Sample Test in High Dimensions Using Random Projections, Journal of Computational and Graphical Statistics, 25, 954-970 (2016) · doi:10.1080/10618600.2015.1062771 |
| [21] | Thulin, M., “A High-Dimensional Two-Sample Test for the Mean Using Random Subspaces, Computational Statistics & Data Analysis, 74, 26-38 (2014) · Zbl 1506.62177 |
| [22] | Tilton, S. C.; Karin, N. J.; Webb-Robertson, B.-J. M.; Waters, K. M.; Mikheev, V.; Lee, K. M.; Corley, R. A.; Pounds, J. G.; Bigelow, D. J., “Impaired Transcriptional Response of the Murine Heart to Cigarette Smoke in the Setting of High Fat Diet and Obesity, Chemical Research in Toxicology, 26, 1034-1042 (2013) · doi:10.1021/tx400078b |
| [23] | Wang, R.; Xu, X., “On Two-Sample Mean Tests Under Spiked Covariances, Journal of Multivariate Analysis, 167, 225-249 (2018) · Zbl 1398.62147 · doi:10.1016/j.jmva.2018.05.004 |
| [24] | Wang, W.; Fan, J., “Asymptotics of Empirical Eigenstructure for High Dimensional Spiked Covariance, The Annals of Statistics, 45, 1342 (2017) · Zbl 1373.62299 · doi:10.1214/16-AOS1487 |
| [25] | Xu, G.; Lin, L.; Wei, P.; Pan, W., “An Adaptive Two-Sample Test for High-Dimensional Means, Biometrika, 103, 609-624 (2016) · Zbl 1506.62314 · doi:10.1093/biomet/asw029 |
| [26] | Xue, K.; Yao, F., “Distribution and Correlation-Free Two-Sample Test of High-Dimensional Means, The Annals of Statistics, 48, 1304-1328 (2020) · Zbl 1454.62157 · doi:10.1214/19-AOS1848 |
| [27] | Zhang, J.-T.; Guo, J.; Zhou, B.; Cheng, M.-Y., “A Simple Two-Sample Test in High Dimensions Based on l^2-Norm, Journal of the American Statistical Association, 115, 1011-1027 (2020) · Zbl 1445.62123 · doi:10.1080/01621459.2019.1604366 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
