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Title: Some geometric approaches to parameter estimation
Author: Hitchcock, David
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1992
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The major part of this thesis is concerned with some geometric aspects of a parameteric statistical problem. In chapter 1 we show how to assign structure to the parameter space by turning it into a Riemannian Manifold. This is achieved by obtaining a metric from the model and the observations in a natural way. We also show how the standard idea of interest and nuisance parameters fits into this context. In chapter 2 the gradient log-likelihood vector field is introduced and a natural diffusion process is put on the parameter space with this vector field as drift. Some properties of this diffusion are investigated including the relationship with the original statistical problem. A method for creating a diffusion on the interest parameter space is exhibited. Chapter 3 considers the case when the nuisance parameters are incidental, ie increase with number of observations. In cases where an optimum method exists for such problems, the method of chapter 2 is equivalent for the right choice of metric. The method is also applied to more general cases and some of the problems that arise are explored. Chapter 4 consists mainly of examples of which the mixture model is probably the most interesting. Chapter 5 is somewhat disconnected from chapters 1-4 and considers observing a (continuous time) parameter dependent stochastic process at discrete time points. The actual likelihood function crucial to the analysis in chapters 1-4 cannot be calculated explicitly so an alternative approach based on martingale techniques is presented. Chapter 6 is self-contained and presents a theorem on the detection of a signal when corrupted by white noise. The likelihood approach used does not appear in previous papers on the subject and leads to a sharper result.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available