IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v66y2014i1p33-61.html
   My bibliography  Save this article

A $$U$$ -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting

Author

Listed:
  • M. Ahmad

Abstract

A two-sample test statistic is presented for testing the equality of mean vectors when the dimension, $$p$$ , exceeds the sample sizes, $$n_i,\; i=1, 2$$ , and the distributions are not necessarily normal. Under mild assumptions on the traces of the covariance matrices, the statistic is shown to be asymptotically Chi-square distributed when $$n_i, p \rightarrow \infty $$ . However, the validity of the test statistic when $$p$$ is fixed but large, including $$p > n_i$$ , and when the distributions are multivariate normal, is shown as special cases. This two-sample Chi-square approximation helps us establish the validity of Box’s approximation for high-dimensional and non-normal data to a two-sample setup, valid even under Behrens–Fisher setting. The limiting Chi-square distribution of the statistic is obtained using the asymptotic theory of degenerate $$U$$ -statistics, and using a result from classical asymptotic theory, it is further extended to an approximate normal distribution. Both independent and paired-sample cases are considered. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Suggested Citation

  • M. Ahmad, 2014. "A $$U$$ -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(1), pages 33-61, February.
    • Handle: RePEc:spr:aistmt:v:66:y:2014:i:1:p:33-61
    DOI: 10.1007/s10463-013-0404-2
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10463-013-0404-2
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10463-013-0404-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Neuhaus, Georg, 1977. "Functional limit theorems for U-statistics in the degenerate case," Journal of Multivariate Analysis, Elsevier, vol. 7(3), pages 424-439, September.
    2. Rauf Ahmad, M. & Werner, C. & Brunner, E., 2008. "Analysis of high-dimensional repeated measures designs: The one sample case," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 416-427, December.
    3. Srivastava, Muni S., 2009. "A test for the mean vector with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 518-532, March.
    4. 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.
    5. Holzmann, Hajo & Koch, Susanne & Min, Aleksey, 2004. "Almost sure limit theorems for U-statistics," Statistics & Probability Letters, Elsevier, vol. 69(3), pages 261-269, September.
    6. Neumeyer, Natalie, 2004. "A central limit theorem for two-sample U-processes," Statistics & Probability Letters, Elsevier, vol. 67(1), pages 73-85, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zongliang Hu & Tiejun Tong & Marc G. Genton, 2019. "Diagonal likelihood ratio test for equality of mean vectors in high‐dimensional data," Biometrics, The International Biometric Society, vol. 75(1), pages 256-267, March.
    2. Liu, Jiamin & Ma, Shuangge & Xu, Wangli & Zhu, Liping, 2022. "A generalized Wilcoxon–Mann–Whitney type test for multivariate data through pairwise distance," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    3. M. Rauf Ahmad, 2019. "A unified approach to testing mean vectors with large dimensions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(4), pages 593-618, December.
    4. Rauf Ahmad, M. & Pavlenko, Tatjana, 2018. "A U-classifier for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 269-283.
    5. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    6. Rauf Ahmad, M., 2019. "A significance test of the RV coefficient in high dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 116-130.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Reza Modarres, 2024. "Hotelling $$T^2$$ T 2 test in high dimensions with application to Wilks outlier method," Statistical Papers, Springer, vol. 65(8), pages 5203-5218, October.
    2. Zhou, Bu & Guo, Jia, 2017. "A note on the unbiased estimator of Σ2," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 141-146.
    3. Saha, Enakshi & Sarkar, Soham & Ghosh, Anil K., 2017. "Some high-dimensional one-sample tests based on functions of interpoint distances," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 83-95.
    4. Jiang Hu & Zhidong Bai & Chen Wang & Wei Wang, 2017. "On testing the equality of high dimensional mean vectors with unequal covariance matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 365-387, April.
    5. Biswas, Munmun & Ghosh, Anil K., 2014. "A nonparametric two-sample test applicable to high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 160-171.
    6. Dong, Kai & Pang, Herbert & Tong, Tiejun & Genton, Marc G., 2016. "Shrinkage-based diagonal Hotelling’s tests for high-dimensional small sample size data," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 127-142.
    7. Shin-ichi Tsukada, 2019. "High dimensional two-sample test based on the inter-point distance," Computational Statistics, Springer, vol. 34(2), pages 599-615, June.
    8. Yin, Yanqing, 2021. "Test for high-dimensional mean vector under missing observations," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    9. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    10. Cai, T. Tony & Xia, Yin, 2014. "High-dimensional sparse MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 174-196.
    11. Amanda Plunkett & Junyong Park, 2019. "Two-sample test for sparse high-dimensional multinomial distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 804-826, September.
    12. Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2018. "Hotelling’s T2 in separable Hilbert spaces," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 284-305.
    13. Yuanyuan Jiang & Xingzhong Xu, 2022. "A Two-Sample Test of High Dimensional Means Based on Posterior Bayes Factor," Mathematics, MDPI, vol. 10(10), pages 1-23, May.
    14. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.
    15. Huang, Yuan & Li, Changcheng & Li, Runze & Yang, Songshan, 2022. "An overview of tests on high-dimensional means," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    16. Anil K. Ghosh & Munmun Biswas, 2016. "Distribution-free high-dimensional two-sample tests based on discriminating hyperplanes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 525-547, September.
    17. Huiqin Li & Jiang Hu & Zhidong Bai & Yanqing Yin & Kexin Zou, 2017. "Test on the linear combinations of mean vectors in high-dimensional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 188-208, March.
    18. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    19. Zhang, Jin-Ting & Zhu, Tianming, 2022. "A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    20. Xu, Kai & Hao, Xinxin, 2019. "A nonparametric test for block-diagonal covariance structure in high dimension and small samples," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 551-567.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aistmt:v:66:y:2014:i:1:p:33-61. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.