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Some aspects of stochastic modelling

Taylor, James Mackenzie

Description

This thesis is mainly concerned with methods for the comparison of distribution functions which depend on the whole of the distribution functions, as against those methods which depend on the comparison of derived statistics. The main methods considered are partial and pre-orderings (Chapter 2) and probability metrics (Chapter 4). Chapter 0 contains a general discussion of mathematical and stochastic modelling. The aim of this chapter is to introduce the concept of the structure of a...[Show more]

dc.contributor.authorTaylor, James Mackenzie
dc.date.accessioned2017-11-06T01:24:21Z
dc.date.available2017-11-06T01:24:21Z
dc.date.copyright1984
dc.identifier.otherb1063306
dc.identifier.urihttp://hdl.handle.net/1885/133195
dc.description.abstractThis thesis is mainly concerned with methods for the comparison of distribution functions which depend on the whole of the distribution functions, as against those methods which depend on the comparison of derived statistics. The main methods considered are partial and pre-orderings (Chapter 2) and probability metrics (Chapter 4). Chapter 0 contains a general discussion of mathematical and stochastic modelling. The aim of this chapter is to introduce the concept of the structure of a stochastic model, which is fundamental to the notion of comparability of stochastic models. Chapter 1 is introductory. Chapter 2 considers the comparison of distribution functions via partial orderings. Various partial and pre-orderings are described and their properties studied. In particular, a survey of comparisons for certain common distribution functions is given. In Chapter 3 it is shown how the orderings introduced in Chapter 2 lead to definitions of monotonicity and comparability for stochastic models. Examples are given for Markov chains, martingales, renewal processes and queueing processes. Proofs of results of Daley (1968), Kalmykov (1962) and others on the monotonicity of Markov chains are shown to be elementary. Chapter 4 deals with the use of metrics to compare distribution functions. Some of the more common probability metrics are listed. New bounds on the supremum (uniform) metric are derived for non-negative integer-valued random variables. Probability metrics are fundamental to the notion of the stability of stochastic models, and this is considered in Chapter 5. In Chapter 6 monotonicity properties are used to investigate the behaviour of a class of branching processes which allow for interaction between male and female in the production of offspring. Sufficient conditions for certain extinction and non-certain extinction are given for the general model, whilst for models with superadditive mating functions necessary and sufficient conditions are given for almost sure extinction. Comparison of models which allow for sexual reproduction with ordinary Galton-Watson branching processes shows that, at least for small populations, there is a significant difference in both the probabilities of extinction and the rates of growth. These are investigated numerically using a truncation technique. Theoretical bounds are given for the error in the calculated values of the extinction probabilities.
dc.format.extentv, [ii], 214 leaves
dc.language.isoen
dc.subject.lcshDistribution (Probability theory)
dc.subject.lcshStochastic systems
dc.subject.lcshMarkov processes
dc.titleSome aspects of stochastic modelling
dc.typeThesis (PhD)
local.contributor.supervisorDaley, D.J.
dcterms.valid1984
local.description.notesThesis (Ph.D.)--Australian National University, 1984. This thesis has been made available through exception 200AB to the Copyright Act.
local.type.degreeDoctor of Philosophy (PhD)
dc.date.issued1984
local.contributor.affiliationDepartment of Statistics, Research School of Social Sciences, The Australian National University
local.identifier.doi10.25911/5d723b9473a6f
dc.date.updated2017-10-20T04:19:15Z
local.mintdoimint
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