This thesis is primarily concerned with the investigation of asymptotic
properties of the maximum likelihood estimate (MLE) of parameters of a
stochastic process. These asymptotic properties are related to martingale
limit theory by recognizing the (known) fact that, under certain regularity
conditions, the derivative of the logarithm of the likelihood function is a
martingale. To this end, part of the thesis is devoted to using or
developing martingale limit theory to provide conditions for the consistency
and/or asymptotic normality of the MLE. Thus, Chapter 1 is concerned with
the martingale limit theory, while the remaining chapters look at its
application to three broad types of stochastic processes. Chapter 2 extends
the classical development of asymptotic theory of MLE’s (a la Cramer [1])
to stochastic processes which, basically, behave in a non-explosive way and
for which non-random norming sequences can be used. In this chapter we also
introduce a generalization of Fisher's measure of information to the
stochastic process situation. Chapter 3 deals with the theory for general
processes and develops the notion of "conditional" exponential families of
processes, as well as establishing the importance of using random norming
sequences. In Chapter 4 we consider the asymptotic theory of maximum
likelihood estimation for continuous time processes and establish results
which are analogous to those for discrete time processes. In each of these
chapters many applications are considered in an attempt to show how known
and new results fit into the general framework of estimation for stochastic
processes.
In Appendix B, a report on the use of the empirical characteristic
function in inference is included in order to indicate how one might deal
with situations where the likelihood is intractable
