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.
Behrens-Fisher problem General linear hypothesis Profile analysis U-statistics
AMS Subject Classification
This is a preview of subscription content, log in to check access.
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
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
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
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
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
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
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
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
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
Davis, C. S. 2002. Statistical methods for the analysis of repeated measurements. New York, NY: Springer.zbMATHGoogle Scholar