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.

In structural equation models, researchers often examine two types of causal effects: direct and indirect effects. Direct effects involve variables that "directly" influence other variables, whereas indirect effects are transmitted via intervening variables. While researchers have paid considerable attention to the distribution of sample direct effects, the distribution of sample indirect effects has only recently been considered. Using the (delta) method (Rao, 1973), Sobel (1982) derived the asymptotic distribution for estimators of indirect effects in recursive systems. Sobel (1986) then derived the asymptotic distribution for estimators of total indirect effects in the Joreskog covariance structure model (Joreskog, 1977). This study examined the applicability of the large sample theory described by Sobel (1986) in small samples. Monte Carlo methods were used to evaluate the behavior of estimated total indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models were used in the analysis. Model 1 was a nonrecursive model with latent variables, feedback, and functional constraints among the effects (Duncan, Haller, & Portes, 1968; Sobel, 1986). Model 2 was a recursive model with observable variables (Duncan, Featherman, & Duncan, 1972). In addition, variations in these models were studied by randomly increasing and decreasing model parameters. The principal findings of the study suggest certain guidelines for researchers who use Sobel's procedures to evaluate total indirect effects in structural equation models. In order for the behavior of the estimates to approximate the asymptotic properties, sample sizes of 400 or more are indicated for nonrecursive systems similar to Model 1, and for recursive systems such as Model 2, sample sizes of 200 or more are suggested. At these sample sizes, researchers can expect sample indirect effects to be accurate point estimators, and confidence intervals for the effects to behave as theory predicts. A caveat to the above guidelines is that, when the total indirect effects are "small" in magnitude, relative to the scale of the model, convergence to the asymptotic properties appears to be very slow. Under these conditions, sampling distributions for the "smaller" valued estimates were positively skewed. This caused estimates to be significantly different from true values, and confidence intervals to behave contrary to theoretical expectations.

by Kwong-hon Ho. / Bibliography: leaves 51-53 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

by Ying-wang Wong. / Year shown on spine: 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 41-44). / Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Multiple Comparison Procedures --- p.1 / Chapter 1.2 --- Familywise Error Rate --- p.3 / Chapter 1.3 --- One-step Procedures Versus Stepwise Procedures --- p.4 / Chapter 1.4 --- Pairwise Multiple Comparisons --- p.5 / Chapter 1.5 --- Pairwise Multiple Comparisons in Two-Way Designs --- p.6 / Chapter 1.6 --- Objectives --- p.8 / Chapter 2. --- Pairwise Multiple Comparisons in One-Way Designs with Covariates --- p.9 / Chapter 2.1 --- The General ANCOVA Model --- p.9 / Chapter 2.2 --- Pairwise Comparisons --- p.12 / Chapter 3. --- Pairwise Comparisons in Two-Way Layout with Covariates --- p.15 / Chapter 3.1 --- The Model --- p.15 / Chapter 3.2 --- The Test Statistics --- p.16 / Chapter 3.3 --- Computation of Upper Percentage Points --- p.17 / Chapter 3.4 --- Approximation Procedure --- p.21 / Chapter 3.5 --- Two-Way Layout with One Covariate --- p.21 / Chapter 4. --- Numerical Examples --- p.23 / Appendix A - An Algorithm in Solving Equation (3.2.4) for the value of ta --- p.35 / Appendix B - Evaluation of Multivariate Normal Probabilities --- p.38 / References --- p.41

Project (M.S.)--Brigham Young University. Dept. of Statistics, 2004. / Includes bibliographical references (p. 109-111).

published_or_final_version / Statistics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 84-86).

It has long been presumed that any consistent nonlinear theory of a spin-two field must be generally covariant. Using Wald's consistency criteria, we exhibit classes of nonlinear theories of a spin-two field that do not have general covariance. We consider four alternative formulations of the spin-two equations. As a first example, we consider a conformally invariant theory of a spin-two field coupled to a scalar field. In the next two cases, the usual symmetric rank-two tensor field, γab, is chosen as the potential. In the fourth case, a traceless symmetric rank-two tensor field is used as the potential. We find that consistent nonlinear generalization of these different formulations leads to theories of a spin-two field that are not generally covariant. In particular, we find types of theories which, when interpreted in terms of a metric, are invariant under the infinitesimal gauge transformation γab→γab + ∇ (a∇[symbol omitted]K[symbol omitted]), where Kab is an arbitrary two-form field. In addition, we find classes of theories that are conformally invariant. As a related problem, we compare the types of theories obtained from the nonlinear extension of a divergence- and curl-free vector field when it is described in terms of two of its equivalent formulations. We find that nonlinear extension of the theory is quite different in each case. Moreover, the resulting types of nonlinear theories may not necessarily be equivalent. A similar analysis is carried out for three-dimensional electromagnetism. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

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