Abstract:
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In multivariate statistical analysis, it is a necessity to know the facts regarding the covariance matrix of the data in hand before applying any further analysis. This study focuses on testing hypotheses concerning the covariance matrices of multivariate normal data having the number of variables larger than or equal to the sample size, called high-dimensional data. The two objectives of this study were: first, for one sample data, to develop a test statistic for testing the hypothesis for whether the covariance matrix equals a specified known matrix, called a partially known matrix, and second, for two independent sample data, to develop a test statistic for testing the hypothesis of equality of two covariance matrices of the two independent populations. For the two hypotheses, a classical method such as the likelihood ratio test is commonly used and is well defined when the sample size is larger than the number of variables. The two proposed test statistics T1 (for one sample data) and T2(for two sample data) are proposed for a high-dimensional situation. Both test statistics T1 and T2 are asymptotically normally distributed when the number of variables and the sample size go towards infinity. A simulation study showed that both proposed test statistics and approximately control the nominal significance level and have good powers. The convergences to asymptotic normality of the two statistics were not greatly affected by the change of covariance structures considered in the study (Unstructured, Compound Symmetry, Heterogeneous Compound Symmetry, Simple, Toeplitz, and Variance Component). Furthermore, in the one sample case, the proposed test statistic T1 performed comparably to the test statistics Uj , proposed by Ledoit and Wolf (2002), and Ts1, proposed by Srivastava (2005), for large sample sizes and was more powerful than these two tests for small or moderate sample sizes with a larger or equal number of variables. In the two sample case, the proposed test statistic T2 is as good as the competitive tests ,Tj, and Tsy proposed by Schott (2007), Srivastava (2007b), and Srivastava and Yanagihara (2010), respectively, for large sample sizes and is markedly superior to these competitive tests for small to moderate sample sizes with a larger or equal number of variables for all of the covariance matrix structures considered. Finally, two real datasets regarding human gene expression in colon tissues were also analyzed to illustrate the application of the theoretical results.
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