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A generalized likelihood ratio test for normal mean when p is greater than n

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  • Zhao, Junguang
  • Xu, Xingzhong

Abstract

The problem of testing the population mean vector of high-dimensional multivariate data is considered. Inspired by Roy’s union–intersection test, a generalized high-dimensional likelihood ratio test for the normal population mean vector is proposed. The p-value for the test is obtained by using randomization method, which does not rely on assumptions about the structure of the covariance matrix. An interpretation of the new statistic is given, which does not rely on the normality assumption. Hence the proposed test is also available for non-normal multivariate population. Simulation studies show that the new test offers higher power than other two competing tests when the variables are dependent and performs particularly well for non-normal multivariate population.

Suggested Citation

  • Zhao, Junguang & Xu, Xingzhong, 2016. "A generalized likelihood ratio test for normal mean when p is greater than n," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 91-104.
    • Handle: RePEc:eee:csdana:v:99:y:2016:i:c:p:91-104
    DOI: 10.1016/j.csda.2016.01.006
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    1. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    2. Thulin, Måns, 2014. "A high-dimensional two-sample test for the mean using random subspaces," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 26-38.
    3. Jeongyoun Ahn & J. S. Marron, 2010. "The maximal data piling direction for discrimination," Biometrika, Biometrika Trust, vol. 97(1), pages 254-259.
    4. Katayama, Shota & Kano, Yutaka & Srivastava, Muni S., 2013. "Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 410-421.
    5. Chen, Song Xi & Li, Jun & Zhong, Pingshou, 2014. "Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation," MPRA Paper 59815, University Library of Munich, Germany.
    6. Sen, Pranab K. & Tsai, Ming-Tien, 1999. "Two-Stage Likelihood Ratio and Union-Intersection Tests for One-Sided Alternatives Multivariate Mean with Nuisance Dispersion Matrix," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 264-282, February.
    7. Shen, Yanfeng & Lin, Zhengyan & Zhu, Jun, 2011. "Shrinkage-based regularization tests for high-dimensional data with application to gene set analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2221-2233, July.
    8. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2010. "Hypothesis Testing in Econometrics," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 75-104, September.
    9. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    10. T. Tony Cai & Weidong Liu & Yin Xia, 2014. "Two-sample test of high dimensional means under dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 349-372, March.
    11. Jianhua Z. Huang & Naiping Liu & Mohsen Pourahmadi & Linxu Liu, 2006. "Covariance matrix selection and estimation via penalised normal likelihood," Biometrika, Biometrika Trust, vol. 93(1), pages 85-98, March.
    12. Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    13. Liang, Jiajuan & Tang, Man-Lai, 2009. "Generalized F-tests for the multivariate normal mean," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1177-1190, February.
    14. Furrer, Reinhard & Bengtsson, Thomas, 2007. "Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 227-255, February.
    15. Mudholkar, Govind S. & Davidson, Michael L. & Subbaiah, Perla, 1974. "A note on the Union-Intersection character of some MANOVA procedures," Journal of Multivariate Analysis, Elsevier, vol. 4(4), pages 486-493, December.
    16. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
    17. Srivastava, Muni S. & Du, Meng, 2008. "A test for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 386-402, March.
    18. Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
    19. Srivastava, Muni S. & Katayama, Shota & Kano, Yutaka, 2013. "A two sample test in high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 349-358.
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    Cited by:

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    4. Zhao, Li & Xu, Xingzhong, 2017. "Generalized canonical correlation variables improved estimation in high dimensional seemingly unrelated regression models," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 119-126.
    5. Ouyang, Yanyan & Liu, Jiamin & Tong, Tiejun & Xu, Wangli, 2022. "A rank-based high-dimensional test for equality of mean vectors," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).

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