Date of Award

8-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Kiessler, Peter C.

Committee Member

Gallagher , Colin

Committee Member

Fralix , Brian

Abstract

This dissertation describes the research that we have done concerning
reversible Markov chains. We first present definitions for what it means
for a Markov chain to be reversible. We then give applications of where
reversible Markov chains are used and give a brief history of Markov chain
inference. Finally, two journal articles are found in the paper, one that
is already published and another which is currently being submitted.
The first article examines estimation of the one-step-ahead
transition probabilities in a reversible Markov chain on a countable state
space. A symmetrized moment estimator is proposed that exploits the
reversible structure. Examples are given where the symmetrized estimator
has superior asymptotic properties to those of a naive estimator,
implying that knowledge of reversibility can sometimes improve estimation.
The asymptotic mean and variance of the estimators are quantified. The
results are proven using only elementary results such as the law of large
numbers and the central limit theorem.
The second article introduces two statistics that assess whether
(or not) a sequence sampled from a time-homogeneous Markov chain on a
finite state space is reversible. The test statistics are based on
observed deviations of transition sample counts between each pair of
states in the chain. First, the joint asymptotic normality of these sample
counts is established. This result is then used to construct two
chi-squared-based tests for reversibility. Simulations assess the power
and type one error of the proposed tests.

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