High Dimensional Multivariate Inference Under General Conditions

In this dissertation, we investigate four distinct and interrelated problems for high-dimensional inference of mean vectors in multi-groups. The first problem concerned is the profile analysis of high dimensional repeated measures. We introduce new test statistics and derive its asymptotic distribution under normality for equal as well as unequal covariance cases. Our derivations of the asymptotic distributions mimic that of Central Limit Theorem with some important peculiarities addressed with sufficient rigor. We also derive consistent and unbiased estimators of the asymptotic variances for equal and unequal covariance cases respectively. The second problem considered is the accurate inference for high-dimensional repeated measures in factorial designs as well as any comparisons among the cell means. We derive asymptotic expansion for the null distributions and the quantiles of a suitable test statistic under normality. We also derive the estimator of parameters contained in the approximate distribution with second-order consistency. The most important contribution is high accuracy of the methods, in the sense that p-values are accurate up to the second order in sample size as well as in dimension. The third problem pertains to the high-dimensional inference under non-normality. We relax the commonly imposed dependence conditions which has become a standard assumption in high dimensional inference. With the relaxed conditions, the scope of applicability of the results broadens. The fourth problem investigated pertains to a fully nonparametric rank-based comparison of high-dimensional populations. To develop the theory in this context, we prove a novel result for studying the asymptotic behavior of quadratic forms in ranks. The simulation studies provide evidence that our methods perform reasonably well in the high-dimensional situation. Real data from Electroencephalograph (EEG) study of alcoholic and control subjects is analyzed to illustrate the application of the results