Some applications of nonparametric regression with constrained data

• Main Author: Maatouk, Talal A. H
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2003
• Subjects: Statistics
• ISBN: 0355980827; 9780355980820

Regression is a very important statistical tool. It helps us to understand relationships between variables, to explore various features of a statistical model and to predict its behaviour. Parametric and nonparametric regression models are particularly important in data analysis. This thesis considers some applications on nonparametric regression with constrained data. There are several forms of nonparametric regression. The main object of this thesis is to concentrate on one of these forms and apply it to three data types, which include interval data, compositional data and spherical data. This thesis has seven chapters and four appendices. Chapter 1 starts with an introduction to the area of nonparametric regression. It introduces the general problem and the essential differences between parametric and nonparametric regression. Also, it reveals the importance of nonparametric regression. The last section outlines the main ideas pursued in this thesis. Chapter 2 presents an overview of issues in smoothing. It introduces basic ideas of some of the existing methods of nonparametric regression and properties of the estimated models. The emphasis is on local linear regression. Also, it deals with the methods of optimal choice of the smoothing parameter with particular attention given to the improved Akaike information criterion (AICc) related method. Chapter 3 employs the cross-validation method and (AICc) related method to investigate the effects of using automatic choices of smoothing parameter in tests of functional shape for regression curves. Simulation studies, based on nonparametric regression, have been performed to test the linearity of a regression function, to compare regression curves and to test the monotonicity of a regression function. Results of the simulation studies are reported. They show good performance of the tests. Chapter 4 deals with least squares fitting of interval data. Two models have been considered, interval-valued output observed from interval-valued input or single-valued input. The general form of the models implies the existence of some different cases and three types of intervals. The solutions follow two approaches, parametric and nonparametric, for three types of intervals. The constraint with interval data is the order of the endpoints. Chapter 5 deals with least squares fitting of compositional data. Three models will be considered, single-valued output resulting from compositional input and compositional output observed from single-valued input or compositional input. Also, the solutions follow two approaches, parametric and nonparametric. The constraint of compositional data is that the sum of the components of a composition is equal to one. Chapter 6 deals with spherical data, where the nonparametric approach of regression and density estimation will be considered. Spherical data requires dealing with edge-effects when the smoothing algorithm is designed. Chapter 7 concludes with a brief review of the main findings of the thesis and a discussion of potential future work on some topics. There are four Appendices at the end of the thesis. They contain a list of symbols and abbreviations, detail of some expressions, tables and softwares used in the thesis.