This thesis aims to develop some new and novel methods in advancing multivariate testing and multiple testing for high-dimensional small sample size data. In Chapter 2, we propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared t-statistics rather than a direct summation of those components. More importantly, to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses, we do not need the requirement that the covariance matrices follow a diagonal matrix structure. As a consequence, our proposed test methods are very flexible and readily applicable in practice. Monte Carlo simulations and a real data analysis are also carried out to demonstrate the advantages of the proposed methods. In Chapter 3, we propose a pairwise Hotelling's method for testing high-dimensional mean vectors. The new test statistics make a compromise on whether using all the correlations or completely abandoning them. To achieve the goal, we perform a screening procedure, pick up the paired covariates with strong correlations, and construct a classical Hotelling's statistic for each pair. While for the individual covariates without strong correlations with others, we apply squared t statistics to account for their respective contributions to the multivariate testing problem. As a consequence, our proposed test statistics involve a combination of the collected pairwise Hotelling's test statistics and squared t statistics. The asymptotic normality of our test statistics under the null and local alternative hypotheses are also derived under some regularity conditions. Numerical studies and two real data examples demonstrate the efficacy of our pairwise Hotelling's test. In Chapter 4, we propose a regularized t distribution and also explore its applications in multiple testing. The motivation of this topic dates back to microarray studies, where the expression levels of thousands of genes are measured simultaneously by the microarray technology. To identify genes that are differentially expressed between two or more groups, one needs to conduct hypothesis test for each gene. However, as microarray experiments are often with a small number of replicates, Student's t-tests using the sample means and standard deviations may suffer a low power for detecting differentially expressed genes. To overcome this problem, we first propose a regularized t distribution and derive its statistical properties including the probability density function and the moments. The noncentral regularized t distribution is also introduced for the power analysis. To demonstrate the usefulness of the proposed test, we apply the regularized t distribution to the gene expression detection problem. Simulation studies and two real data examples show that the regularized t-test outperforms the existing tests including Student's t-test and the Bayesian t-test in a wide range of settings, in particular when the sample size is small.

When there are several possible segmentation variables, marketers must investigate the ramifications of their potential interactions. These include their mutual association, the identification of the best (the distinguished) segmentation variable and its predictability by a set of descriptor variables, and the structure of the multivariate system(s) obtained from the segmentation and descriptor variables. This procedure has been defined as segment congruence analysis (SCA). This study utilizes the information theoretic and the log-linear/logit approaches to address a variety of research questions in segment congruence analysis. It is shown that the information theoretic approach expands the scope of SCA and offers some advantages over traditional methods. Data obtained from a survey conducted by the Bonneville Power Administration (BPA) and Northwest utilities is used to demonstrate the information theoretic and the log-linear/logit approaches and compare these two methods. The survey was designed to obtain information on energy consumption habits, attitudes toward selected energy issues, and the conservation measures utilized by the residents in the Pacific Northwest. The analyses are performed in two distinct phases. Phase I includes assessment of mutual association among segmentation variables and four methods (based on different information theoretic functions) for identifying candidates for the distinguished variable. Phase II addresses the selection and analysis of the distinguished variable. This variable is selected either a priori or by assessment of its predictability from (segmentation or exogenous) descriptor variables. The relations between the distinguished variable and the descriptor variables are further analyzed by examining the predictability issue in greater detail and by evaluating structural models of the multivariate systems. The methodological conclusions of this study are that the information theoretic and log-linear methods have deep similarities. The analyses produced intuitively plausible results. In Phase I, energy related awareness, behavior, perceptions, attitudes, and electricity consumption were identified as candidate segmentation variables. In Phase II, using exogenous descriptor variables, electricity consumption was selected as the distinguished variable. The analysis of this variable indicated that the demographic factors, type of dwelling, and geoclimatic environment are among the most important determinants of electricity consumption.

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Thesis (M. Sc.)--Carleton University, 2005. / Includes bibliographical references (p. 62-67). Also available in electronic format on the Internet.

Thesis (M.S.)--University of Michigan, 1963.

Thesis (M.S.)--University of Michigan, 1963.

<p>Canonical Correlation Analysis (CCA) is one of the multivariate statistical methods that can be used to find relationship between two sets of variables. I highlighted challenges in analyzing high-dimensional data with CCA. Recently, Sparse CCA (SCCA) methods have been proposed to identify sparse linear combinations of two sets of variables with maximal correlation in the context of high-dimensional data. In my thesis, I compared three different SCCA approaches. I evaluated the three approaches as well as the classical CCA on simulated datasets and illustrated the methods with publicly available genomic and proteomic datasets.</p> / Master of Science (MSc)