Graphics and Inference in Nonparametric Modelling

Young, Stuart Gordon (1996) Graphics and Inference in Nonparametric Modelling. PhD thesis, University of Glasgow.

Abstract

This thesis is concerned with statistical modelling techniques which involve nonpara- metric smoothing. Its principal aim is to introduce a number of graphical methods of assessing nonparametric models, and also examine the important area of formal inference in the nonparametric setting. Chapter 1 gives a broad introduction to the ideas and methods contained in this thesis, and includes a description of kernel smoothing, and some key results. Chapter 2 leads off with a general treatment of three-dimensional data, and a method of displaying such data, by including a surface, relating to a model of interest. This surface can bring out the three-dimensional structure by providing a helpful visual reference. This basic idea is applied to a number of areas, such as regression, survival data, principal component analysis, and linear discrimination. Nonparametric modelling is introduced towards the end of this chapter, when three-dimensional density estimation is considered, and a contouring procedure examined. In the area of nonparametric smoothing, whether density estimation or regression, much of the literature deals with the selection of a smoothing parameter. A common feature of many of the nonparametric techniques introduced in this thesis is that the influence of the smoothing parameter is reduced. With this often contentious issue assuming less importance, more emphasis can be placed on meaningful interpretations of the data. The three-dimensional contouring procedure discussed here is the first example of this approach. Exploring the nonparametric theme more fully, Chapter 3 deals with nonparametric regression, and in particular, graphical methods of comparing several curves. This comparison is achieved by means of a "reference band", which, while not removing the need for a formal test, can provide a useful visual interpretation of a model. Reference bands for equality are derived and explored in a variety of settings. Reference bands for parallelism are also derived for nonparametric regression models. The subject of bias, so often a concern in nonparametric modelling, is eliminated in this context by appropriate choice of smoother. The important topic of inference is addressed in Chapter 4, which introduces tests for a nonparametric analysis of covariance model, which provide formal means of analysis to accompany some of the graphical methods introduced in Chapter 3. Tests for equality and parallelism are derived, and their power assessed via a simulation study. Chapter 5 extends the ideas of Chapter 4 by applying them to binary response data, and a test for equality is introduced for this case. The approach here is necessarily different from that in Chapter 4, and involves deriving the mean and variance of the test statistic exactly, in order to permit a null distribution to be found. An example from Chapter 3 is re-visited here, and the formal test applied, to accompany the reference band produced previously. As in Chapter 4, the size and power of the new test is investigated via a simulation study. Competing null distributions are also assessed for their suitability. Finally, Chapter 6 widens the scope of the thesis, by considering more general models in several covariates. Graphical methods and formal tests, which involve looking for patterns in residuals, are considered for semiparametric models, containing both parametric and nonparametric components. Inference in bivariate smoothing is also addressed, with a particular application in image analysis. This is extended to a general test for comparing smooth surfaces, using the principles of Chapter 4. The thesis closes with a discussion of the results presented, and what further areas could be investigated.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Adviser: Adrian Bowman
• Keywords: Statistics
• Date of Award: 1996
• Depositing User: Enlighten Team
• Unique ID: glathesis:1996-76484
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 19 Nov 2019 14:17
• Last Modified: 19 Nov 2019 14:17
• URI: https://theses.gla.ac.uk/id/eprint/76484