Projection Test with Sparse Optimal Direction for High-Dimensional One Sample Mean Problem

  • Wanjun LiuEmail author
  • Runze Li


Testing whether the mean vector from some population is zero or not is a fundamental problem in statistics. In the high-dimensional regime, where the dimension of data p is greater than the sample size n, traditional methods such as Hotelling’s \(T^2\) test cannot be directly applied. One can project the high-dimensional vector onto a space of low dimension and then traditional methods can be applied. In this paper, we propose a projection test based on a new estimation of the optimal projection direction \(\varSigma ^{-1}\mu \). Under the assumption that the optimal projection \(\varSigma ^{-1}\mu \) is sparse, we use a regularized quadratic programming with nonconvex penalty and linear constraint to estimate it. Simulation studies and real data analysis are conducted to examine the finite sample performance of different tests in terms of type I error and power.



This work was supported by a NSF grant DMS 1820702 and a NIDA, NIH grant P50 DA039838. The content is solely the responsibility of the authors and does not necessarily represent the official views of NSF, NIH or NIDA.


  1. 1.
    Bai, Z., Saranadasa, H.: Effect of high dimension: by an example of a two sample problem. Statistica Sinica 6, 311–329 (1996)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cai, T., Liu, W.: Adaptive thresholding for sparse covariance matrix estimation. J. Am. Stat. Assoc. 106(494), 672–684 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cai, T., Liu, W., Luo, X.: A constrained \(\ell _1\) minimization approach to sparse precision matrix estimation. J. Am. Stat. Assoc. 106(494), 594–607 (2011)zbMATHGoogle Scholar
  4. 4.
    Cai, T., Liu, W., Xia, Y.: Two-sample test of high dimensional means under dependence. J. R. Stat. Soc. Series B (Statistical Methodology) 76(2), 349–372 (2014)MathSciNetGoogle Scholar
  5. 5.
    Chen, S.X., Li, J., Zhong, P.-S.: Two-sample and ANOVA tests for high dimensional means. Ann. Stat. 47(3), 1443–1474 (2019)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, S.X., Qin, Y.-L.: A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Stat. 38(2), 808–835 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Consortium, W. T. C. C: Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls. Nature 447(7145), 661 (2007)Google Scholar
  8. 8.
    Dempster, A.P.: A high dimensional two sample significance test. Ann. Math. Stat. 29(4), 995–1010 (1958)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fan, J., Xue, L., Zou, H.: Strong oracle optimality of folded concave penalized estimation. Ann. Stat. 42(3), 819 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fang, K.-T., Kotz, S., Ng, K.: Symmetric Multivariate and Related Distributions. Chapman and Hall (1990)Google Scholar
  12. 12.
    Fang, K.-T., Zhang, Y.-T.: Generalized Multivariate Analysis. Science Press, Springer-Verlag, Beijing (1990)zbMATHGoogle Scholar
  13. 13.
    Hotelling, H.: The generalization of student’s ratio. Ann. Math. Stat. 2(3), 360–378 (1931)zbMATHGoogle Scholar
  14. 14.
    Lauter, J.: Exact \(t\) and \(F\) tests for analyzing studies with multiple endpoints. Biometrics 52(3), 964–970 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, R., Huang, Y., Wang, L., Xu, C.: Projection Test for High-dimensional Mean Vectors with Optimal Direction (2015)Google Scholar
  16. 16.
    Lopes, M., Jacob, L., Wainwright, M.J.: A more powerful two-sample test in high dimensions using random projection. In: Advances in Neural Information Processing Systems, pp. 1206–1214 (2011)Google Scholar
  17. 17.
    Pan, G., Zhou, W.: Central limit theorem for hotellings \(T^2\) statistic under large dimension. Ann. Appl. Probab. 21(5), 1860–1910 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Percival, C.J., Huang, Y., Jabs, E.W., Li, R., Richtsmeier, J.T.: Embryonic craniofacial bone volume and bone mineral density in fgfr2+/p253r and nonmutant mice. Dev. Dyn. 243(4), 541–551 (2014)Google Scholar
  19. 19.
    Srivastava, M.S., Du, M.: A test for the mean vector with fewer observations than the dimension. J. Multivar. Anal. 99(3), 386–402 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wang, L., Kim, Y., Li, R.: Calibrating nonconvex penalized regression in ultra-high dimension. Ann. Stat. 41(5), 2505 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Xu, G., Lin, L., Wei, P., Pan, W.: An adaptive two-sample test for high-dimensional means. Biometrika 103(3), 609–624 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zou, H., Li, R.: One-step sparse estimates in nonconcave penalized likelihood models. Ann. Stat. 36(4), 1509 (2008)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsPennsylvania State UniversityUniversity ParkUSA

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