Date of Award

12-2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Sun, Xiaoqian

Committee Member

Gallagher , Colin M

Committee Member

Kulasekera , K.B.

Committee Member

Park , Chanseok

Abstract

Many applications of modern science involve a large number of parameters. In
many cases, the number of parameters, p, exceeds the number of observations,
N. Classical multivariate statistics are based on the assumption that the
number of parameters is fixed and the number of observations is large. Many of
the classical techniques perform poorly, or are degenerate, in high-dimensional
situations.
In this work, we discuss and develop statistical methods for inference of
data in which the number of parameters exceeds the number of observations.
Specifically we look at the problems of hypothesis testing regarding and the
estimation of the covariance matrix.
A new test statistic is developed for testing the hypothesis that the
covariance matrix is proportional to the identity. Simulations show this newly
defined test is asymptotically comparable to those in the literature.
Furthermore, it appears to perform better than those in the literature under
certain alternative hypotheses.
A new set of Stein-type shrinkage estimators are introduced for estimating
the covariance matrix in large-dimensions. Simulations show that under the
assumption of normality of the data, the new estimators are comparable to those
in the literature. Simulations also indicate the new estimators perform better
than those in the literature in cases of extreme high-dimensions. A data
analysis of DNA microarray data also appears to confirm our results of improved
performance in the case of extreme high-dimensionality.

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