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Theory and Methods

A Simple Two-Sample Test in High Dimensions Based on L2-Norm

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Pages 1011-1027 | Received 29 May 2018, Accepted 25 Mar 2019, Published online: 30 May 2019

References

  • Anderson, T. W. (2003), An Introduction to Multivariate Statistical Analysis (3rd ed.), New York: Wiley.
  • Aoshima, M., and Yata, K. (2018), “Two-Sample Tests for High-Dimension, Strongly Spiked Eigenvalue Models,” Statistica Sinica, 28, 43–62.
  • Bai, Z., and Saranadasa, H. (1996), “Effect of High Dimension: By an Example of a Two Sample Problem,” Statistica Sinica, 6, 311–329.
  • Box, G. E. P. (1954), “Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification,” The Annals of Mathematical Statistics, 25, 290–302. DOI: 10.1214/aoms/1177728786.
  • Cai, T. T., Liu, W., and Xia, Y. (2014), “Two-Sample Test of High Dimensional Means Under Dependence,” Journal of the Royal Statistical Society, Series B, 76, 349–372. DOI: 10.1111/rssb.12034.
  • Chen, S., Zhang, L., and Zhong, P. (2010), “Tests for High-Dimensional Covariance Matrices,” Journal of the American Statistical Association, 105, 810–819. DOI: 10.1198/jasa.2010.tm09560.
  • Chen, S. X., and Qin, Y.-L. (2010), “A Two-Sample Test for High-Dimensional Data With Applications to Gene-Set Testing,” The Annals of Statistics, 38, 808–835. DOI: 10.1214/09-AOS716.
  • Cheng, M.-Y., and Wu, H.-T. (2013), “Local Linear Regression on Manifolds and Its Geometric Interpretation,” Journal of the American Statistical Association, 108, 1421–1434. DOI: 10.1080/01621459.2013.827984.
  • Chuang, L.-L., and Shih, Y.-S. (2012), “Approximated Distributions of the Weighted Sum of Correlated Chi-Squared Random Variables,” Journal of Statistical Planning and Inference, 142, 457–472. DOI: 10.1016/j.jspi.2011.08.004.
  • Cook, R. D., and Li, B. (2002), “Dimension Reduction for the Conditional Mean in Regression,” The Annals of Statistics, 30, 455–474. DOI: 10.1214/aos/1021379861.
  • Cui, H., Li, R., and Zhong, W. (2015), “Model-Free Feature Screening for Ultra-High Dimensional Discriminant Analysis,” Journal of the American Statistical Association, 110, 630–641. DOI: 10.1080/01621459.2014.920256.
  • Dempster, A. P. (1958), “A High Dimensional Two Sample Significance Test,” The Annals of Mathematical Statistics, 29, 995–1010. DOI: 10.1214/aoms/1177706437.
  • Dempster, A. P. (1960), “A Significance Test for the Separation of Two Highly Multivariate Small Samples,” Biometrics, 16, 41–50. DOI: 10.2307/2527954.
  • Fan, J., and Fan, Y. (2008), “High Dimensional Classification Using Features Annealed Independence Rules,” The Annals of Statistics, 36, 2605–2637. DOI: 10.1214/07-AOS504.
  • Fan, J., Feng, Y., and Tong, X. (2012), “A Road to Classification in High Dimensional Space: The Regularized Optimal Affine Discriminant,” Journal of the Royal Statistical Society, Series B, 74, 745–771. DOI: 10.1111/j.1467-9868.2012.01029.x.
  • Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360. DOI: 10.1198/016214501753382273.
  • Fan, J., and Lv, J. (2008), “Sure Independence Screening for Ultrahigh Dimensional Feature Space,” Journal of the Royal Statistical Society, Series B, 70, 849–911. DOI: 10.1111/j.1467-9868.2008.00674.x.
  • Fan, J., and Peng, H. (2004), “Nonconcave Penalized Likelihood With a Diverging Number of Parameters,” The Annals of Statistics, 32, 928–961. DOI: 10.1214/009053604000000256.
  • Fan, J., and Song, R. (2010), “Sure Independence Screening in Generalized Linear Models With NP-Dimensionality,” The Annals of Statistics, 38, 3567–3604. DOI: 10.1214/10-AOS798.
  • Fan, Y., Kong, Y., Li, D., and Zheng, Z. (2015), “Innovated Interaction Screening for High-Dimensional Nonlinear Classification,” The Annals of Statistics, 43, 1243–1272. DOI: 10.1214/14-AOS1308.
  • Feiveson, A. H., and Delaney, F. C. (1968), The Distribution and Properties of a Weighted Sum of Chi Squares (Vol. 4575), Washington, DC: National Aeronautics and Space Administration.
  • Feng, L., Zou, C., Wang, Z., and Zhu, L. (2015), “Two Sample Behrens–Fisher Problem for High-Dimensional Data,” Statistica Sinica, 25, 1297–1312.
  • Gregory, K. B., Carroll, R. J., Baladandayuthapani, V., and Lahiri, S. N. (2015), “A Two-Sample Test for Equality of Means in High Dimension,” Journal of the American Statistical Association, 110, 837–849. DOI: 10.1080/01621459.2014.934826.
  • Himeno, T., and Yamada, T. (2014), “Estimations for Some Functions of Covariance Matrix in High Dimension Under Non-Normality and Its Applications,” Journal of Multivariate Analysis, 130, 27–44. DOI: 10.1016/j.jmva.2014.04.020.
  • Hotelling, H. (1931), “The Generalization of Student’s Ratio,” The Annals of Mathematical Statistics, 2, 360–378. DOI: 10.1214/aoms/1177732979.
  • Imhof, J. P. (1961), “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, 48, 419–426. DOI: 10.1093/biomet/48.3-4.419.
  • Katayama, S., Kanoa, Y., and Srivastava, M. S. (2013), “Asymptotic Distributions of Some Test Criteria for the Mean Vector With Fewer Observations Than the Dimension,” Journal of Multivariate Analysis, 116, 410–421. DOI: 10.1016/j.jmva.2013.01.008.
  • Li, G., Peng, H., Zhang, J., and Zhu, L. (2012), “Robust Rank Correlation Based Screening,” The Annals of Statistics, 40, 1846–1877. DOI: 10.1214/12-AOS1024.
  • Lv, J., and Fan, Y. (2009), “A Unified Approach to Model Selection and Sparse Recovery Using Regularized Least Squares,” The Annals of Statistics, 37, 3498–3528. DOI: 10.1214/09-AOS683.
  • Pauly, M., Ellenberger, D., and Brunner, E. (2015), “Analysis of High-Dimensional One Group Repeated Measures Designs,” Statistics, 49, 1243–1261. DOI: 10.1080/02331888.2015.1050022.
  • Satterthwaite, F. E. (1941), “Synthesis of Variance,” Psychometrika, 6, 309–316. DOI: 10.1007/BF02288586.
  • Satterthwaite, F. E. (1946), “An Approximate Distribution of Estimates of Variance Components,” Biometrics Bulletin, 2, 110–114. DOI: 10.2307/3002019.
  • Srivastava, M. S., and Du, M. (2008), “A Test for the Mean Vector With Fewer Observations Than the Dimension,” Journal of Multivariate Analysis, 99, 386–402. DOI: 10.1016/j.jmva.2006.11.002.
  • Srivastava, M. S., Katayama, S., and Kano, Y. (2013), “A Two Sample Test in High Dimensional Data,” Journal of Multivariate Analysis, 114, 349–358. DOI: 10.1016/j.jmva.2012.08.014.
  • Srivastava, M. S., and Kubokawa, T. (2013), “Tests for Multivariate Analysis of Variance in High Dimension Under Non-Normality,” Journal of Multivariate Analysis, 115, 204–216. DOI: 10.1016/j.jmva.2012.10.011.
  • Srivastava, R., Li, P., and Ruppert, D. (2015), “RAPTT: An Exact Two-Sample Test in High Dimensions Using Random Projections,” Journal of Computational and Graphical Statistics, 25, 954–970. DOI: 10.1080/10618600.2015.1062771.
  • Tibshirani, R. J. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288. DOI: 10.1111/j.2517-6161.1996.tb02080.x.
  • van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge, UK: Cambridge University Press.
  • Wang, L., Peng, B., and Li, R. (2015), “A High-Dimensional Nonparametric Multivariate Test for Mean Vector,” Journal of the American Statistical Association, 110, 1658–1669. DOI: 10.1080/01621459.2014.988215.
  • Wang, R., Peng, L., and Qi, Y. (2013), “Jackknife Empirical Likelihood Test for Equality of Two High Dimensional Means,” Statistica Sinica, 23, 667–690.
  • Welch, B. L. (1947), “The Generalization of ‘Student’s’ Problem When Several Different Population Variances Are Involved,” Biometrika, 34, 28–35. DOI: 10.2307/2332510.
  • Witten, D., and Tibshirani, R. (2011), “Penalized Classification Using Fisher’s Linear Discriminant,” Journal of the Royal Statistical Society, Series B, 73, 753–772. DOI: 10.1111/j.1467-9868.2011.00783.x.
  • Xia, Y. (2007), “A Constructive Approach to the Estimation of Dimension Reduction Directions,” The Annals of Statistics, 35, 2654–2690. DOI: 10.1214/009053607000000352.
  • Xia, Y. (2008), “A Multiple-Index Model and Dimension Reduction,” Journal of the American Statistical Association, 103, 1631–1640. DOI: 10.1198/016214508000000805.
  • Yamada, T., and Himeno, T. (2015), “Testing Homogeneity of Mean Vectors Under Heteroscedasticity in High-Dimension,” Journal of Multivariate Analysis, 139, 7–27. DOI: 10.1016/j.jmva.2015.02.005.
  • Zhang, J.-T. (2005), “Approximate and Asymptotic Distributions of Chi-Square-Type Mixtures With Applications,” Journal of the American Statistical Association, 100, 273–285. DOI: 10.1198/016214504000000575.
  • Zhang, J.-T., Guo, J., and Zhou, B. (2017), “Linear Hypothesis Testing in High-Dimensional One-Way MANOVA,” Journal of Multivariate Analysis, 155, 200–216. DOI: 10.1016/j.jmva.2017.01.002.
  • Zhang, R., Peng, L., and Wang, R. (2013), “Tests for Covariance Matrix With Fixed or Divergent Dimension,” The Annals of Statistics, 41, 2075–2096. DOI: 10.1214/13-AOS1136.

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