Some problems concerning point processes
Date
1981
Authors
Littlejohn, Roger Philip
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Abstract
In this thesis we discuss a variety of problems concerning
point processes and Markov processes; the dominant themes are the
contamination and thinning of point processes, small sample theory and
the power of tests of certain hypotheses against given alternatives.
In Chapter 2 we review the literature on the superposition and
thinning of point processes, interpreting these operations as types of
contamination. In Chapter 3 we derive the distribution of the serial
correlation coefficients for a finite portion of a renewal process
which has been contaminated by superposed or deleted points, which leads
us to conclude that the serial correlogram is less sensitive to
contamination than had previously been thought (Shiavi and Negin (1973)).
As an ancilliary result we give expressions for the interval moments
for a length-biased sample from a renewal process of finite length.
In Chapter U we study selective interaction models within the
framework of regenerative bivariate point processes (Berman (1978)) for
which we give counting and interval properties. We classify the many
variations of this model and are able to derive simplified expressions
for first and second order properties; we introduce a thinning operation
where the probability that a point is deleted is a function of the
length of the interval preceding it. In Chapter 5 we study the square wave modulated Poisson process, giving its counting and interval
properties; a problem of inference leads us to generalize Fisher's
(1929) g-statistic for the distribution of the largest and second largest
intervals on a circle and the distribution of the maximum periodogram
ordinate. In Chapter 6 we show that the stationary exponential Markov chains
of Tavares (1980) and Gaver and Lewis (1980) are time reversed versions
of one another and obtain a characterization of the exponential
distribution as a converse. In Chapter 7 ve use a deficiency criterion
(Hodges and Lehmann (1970)) based on the Fisher information matrix
to study the role of initial conditions in estimating the parameters of
a Markov chain. Closed form expressions are given for the maximum
likelihood estimators of the parameters of the simple Markov chain when
the initial conditions are used. We conclude that there is a basis for
using one’s knowledge of the initial conditions for small samples when
the chain is likely to remain in one or more states for long periods,
although this involves more complicated computations than when the
likelihood is conditioned on the value of the initial state.
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Thesis (PhD)
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