Some problems concerning point processes

Date

1981

Authors

Littlejohn, Roger Philip

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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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