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Journal of Statistical Theory and Practice

, Volume 9, Issue 3, pp 544–570 | Cite as

U-Tests of General Linear Hypotheses for High-Dimensional Data Under Nonnormality and Heteroscedasticity

  • M. Rauf Ahmad
Article

Abstract

Test statistics are presented for general linear hypotheses, with special focus on the two-sample profile analysis. The statistics are a modification to the classical Hotelling’s T2 statistic, are basically designed for the case when the dimension, p, may exceed the sample sizes, ni, and are valid under the violation of any assumption associated with T2, such as normality, homoscedasticity, or equal sample sizes. Under a few mild assumptions replacing the classical ones, the test statistics are shown to follow a normal limit under both the null and alternative hypothesis. As the test statistics are defined as a linear combination of U-statistics, the limits are correspondingly obtained using the asymptotic theory of degenerate (for null) and nondegenerate (for alternative) U-statistics. Simulation results, under a variety of parameter settings, are used to show the accuracy and robustness of the test statistics. Practical application of the tests is also illustrated using a few real data sets.

Keywords

Behrens-Fisher problem General linear hypothesis Profile analysis U-statistics 

AMS Subject Classification

62415 

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References

  1. Ahmad, M. R. 2014a. Location-invariant and non-invariant tests for large dimensional covariance matrices under normality and non-normality. Working Papers Series, No. 2014:4, Department of Statistics, Uppsala University, Uppsala, Sweden.Google Scholar
  2. Ahmad, M. R. 2014b. A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting. Ann. Inst. Stat. Math., 66(1), 33–61.MathSciNetCrossRefGoogle Scholar
  3. Ahmad, M. R. 2014c. U-tests for general linear hypotheses under non-normality and heteroscedasticity. Technical report, Department of Statistics, Uppsala University, Uppsala, Sweden.Google Scholar
  4. Ahmad, M. R., D. von Rosen, and M. Singull. 2012. A note on mean testing for high-dimensional multivariate data under non-normality. Stat. Neerl., 67(1), 81–99.MathSciNetCrossRefGoogle Scholar
  5. Ahmad, M. R., and T. Yamada. 2013. Testing homogeneity of covariance matrices and multi-sample sphericity for high-dimensional data. Working paper 2013:1, Department of Statistics, Uppsala University, Uppsala, Sweden.Google Scholar
  6. Anderson, N. H., P. Hall, and D. M. Titterington. 1994. Two-sample test statistics for measuring discrepencies between two multivariate probability density functions using kernel based density estimates. J. Multivariate Anal., 50, 41–54.MathSciNetCrossRefGoogle Scholar
  7. Bai, Z., and H. Saranadasa. 1996. Effect of high dimension: By an example of a two sample problem. Stat. Sin., 6, 311–329.MathSciNetzbMATHGoogle Scholar
  8. Broocks, A., T. Meyer, A. George, et al. 1998. Decreased neuroendocrine responses to meta-cholorophenylpiperazine (m-CPP) but normal responses to ipsapirone in marathon runners. Neuropsychopharmacology, 20(2), 150–161.CrossRefGoogle Scholar
  9. Chen, S. X., and Y.-L. Qin. 2010. A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Stat., 38(2), 808–835.MathSciNetCrossRefGoogle Scholar
  10. Davis, C. S. 2002. Statistical methods for the analysis of repeated measurements. New York, NY: Springer.zbMATHGoogle Scholar
  11. Dempster, A. P. 1958. A high dimensional two sample significance test. Ann. Math. Stat., 29(4), 995–1010.MathSciNetCrossRefGoogle Scholar
  12. Hájek, J., Z. Šidák, and P. K. Sen. 1999. Theory of rank tests. San Diego, CA: Academic Press.zbMATHGoogle Scholar
  13. Hoeffding, W. 1948. A class of statistics with asymptotically normal distribution. Ann. Math. Stat., 19, 293–325.MathSciNetCrossRefGoogle Scholar
  14. Jiang, J. 2010. Large sample techniques for statistics. New York, NY: Springer.CrossRefGoogle Scholar
  15. Koroljuk, V. S., and Y. V. Borovskich. 1994. Theory of U-statistics. Dordrecht, The Netherlands: Kluwer Academic Press.CrossRefGoogle Scholar
  16. Lee, A. J. 1990. U-statistics: Theory and practice. Boca Raton, FL: CRC Press.zbMATHGoogle Scholar
  17. Lehmann, E. L. 1999. Elements of large-sample theory. New York, NY: Springer.CrossRefGoogle Scholar
  18. Liese, F., and K.-J. Miescke. 2008. Statistical decision theory: Estimation, testing and selection. New York, NY: Springer.CrossRefGoogle Scholar
  19. Morrison, D. F. 1967. Multivariate statistical methods. New York, NY: McGraw-Hill.zbMATHGoogle Scholar
  20. Muirhead, R. J. 2005. Aspects of multivariate statistical theory. New York, NY: Wiley.zbMATHGoogle Scholar
  21. Seber, G. A. F. 1966. The linear hypothesis: A general theory. London, UK: Griffin.zbMATHGoogle Scholar
  22. Seber, G. A. F. 2004. Multivariate observations. New York, NY: Wiley.zbMATHGoogle Scholar
  23. Serfling, R. J. 1980. Approximation theorems of mathematical statistics. Weinheim, Germany: Wiley.CrossRefGoogle Scholar
  24. Srivastava, M. S. 2007. Multivariate theory for analyzing high dimensional data. J. Jpn. Stat. Assoc., 37, 53–86.MathSciNetCrossRefGoogle Scholar
  25. van der Vaart, A. W. 1998. Asymptotic statistics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  26. Wolfinger, R. D. 1996. Heterogenous variance: Covariance structures for repeated measures. J. Agric. Biol. Environ. Stat., 1, 205–230.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Department of StatisticsUppsala University (Ekonomikum)UppsalaSweden
  2. 2.Department of Energy and TechnologySwedish University of Agricultural SciencesUppsalaSweden

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