Methods of Inference for Nonparametric Curves and Surfaces

Bock, Mitchum T (1999) Methods of Inference for Nonparametric Curves and Surfaces. PhD thesis, University of Glasgow.

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Nonparametric regression models offer attractive extensions to the familiar approaches of parametric regression. They adapt to departures from standard parametric forms and therefore have the potential to capture features which may otherwise go unnoticed. This property accounts for the large volume of work in the area of estimation of nonparametric models which has emerged over the last two decades. Inferential techniques using nonparametric model fits, however, have not been so quick to develop. This thesis contributes to this area of research by examining the task of assessing covariate effects via comparisons of nonparametric model fits. In particular, the asymptotic and finite sample bias properties of estimates obtained via local linear smoothers are a major consideration and methods of inference which take into account these properties are developed. Chapter 1 introduces and presents an overview of existing methods of estimation and inference amongst nonparametric regression. Chapter 2 focuses on the task of inference by considering the estimation of the error variance in the nonparametric model context. Special attention is given to the development and assessment of difference based estimators in the presence of two covariates. It is shown that difference based estimators are a viable alternative, in terms of accuracy, to standard residual based estimators. Chapters 3 and 4 employ the estimators of Chapter 2 in the development of test procedures which make comparisons amongst a class of bivariate nonparametric regression models. Chapter 3 develops the theoretical properties of several forms of the test statistic, with particular attention given to statistics based on direct comparisons of fitted values. The theory also highlights the role of centred smoothers and equivalent degrees of smoothing when nonparametric model fits are compared. The simulation studies reported in Chapter 4 compare the novel approaches developed in Chapter 3 with standard approaches based on differences in residual sums of squares, i.e. approximate F-tests. The results show that direct comparisons of fitted values offer an improvement in some settings and never perform less favourably in others. The choice of the error variance estimator is shown to be crucial, with different design spaces requiring different estimators. Specific attention is also given to the effect of correlation amongst the covariates on the tests' performances. Chapter 4 closes with an application of the methods to a real data set describing the spatial distribution of sea bed fauna in the Great Barrier Reef. Chapter 5 extends these methods beyond models with two covariates to models with an unlimited number of additive linear terms and a nonparametric component involving at most two covariates. Recent results which derive the asymptotic properties of models of this form show that the favourable properties of local linear regression in the bivariate setting extend to this multidimensional setting. Results of a simulation study are reported and show that there is much to be gained by making a direct comparison of fitted values in conjunction with a careful choice of the estimator of error variance. Chapters 6 and 7 describe applied projects in environmental and medical contexts respectively. Both of the sets of data contain relationships amongst covariates which are best described using nonparametric models. Chapter 6 considers 14 years of water quality monitoring data from the Firth of Clyde, Scotland. Interest lies in describing relationships between pollutants and environmental factors, including long term trends and seasonal patterns. Chapter 7 investigates the relationship between short term dosage of an immunosuppressive drug and the long term outcome of kidney transplantation patients. Chapter 8 concludes with a summary of the main findings of the thesis and a discussion of potential future work in this area. Although progress has been made in the settings considered in the thesis, further extensions are required before nonparametric modelling will achieve its full potential.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Additional Information: Adviser: Adrian Bowman
Keywords: Mathematics
Date of Award: 1999
Depositing User: Enlighten Team
Unique ID: glathesis:1999-76460
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 19 Nov 2019 14:19
Last Modified: 19 Nov 2019 14:19

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