This monograph reviews some of the work that has been done for longitudi nal data in the rapidly expanding field of nonparametric regression. The aim is to give the reader an impression of the basic mathematical tools that have been applied, and also to provide intuition about the methods and applications. Applications to the analysis of longitudinal studies are emphasized to encourage the non-specialist and applied statistician to try these methods out. To facilitate this, FORTRAN programs are provided which carry out some of the procedures described in the text. The emphasis of most research work so far has been on the theoretical aspects of nonparametric regression. It is my hope that these techniques will gain a firm place in the repertoire of applied statisticians who realize the large potential for convincing applications and the need to use these techniques concurrently with parametric regression. This text evolved during a set of lectures given by the author at the Division of Statistics at the University of California, Davis in Fall 1986 and is based on the author's Habilitationsschrift submitted to the University of Marburg in Spring 1985 as well as on published and unpublished work. Completeness is not attempted, neither in the text nor in the references. The following persons have been particularly generous in sharing research or giving advice: Th. Gasser, P. Ihm, Y. P. Mack, V. Mammi tzsch, G . G. Roussas, U. Stadtmuller, W. Stute and R.

Autorentext
Hans-Georg Müller wurde am 27.August 1937 in Suhl in Thüringen geboren. Nach dem Abitur 1955 verließ er die DDR und studierte an den Unis Göttingen und Grenoble in Frankreich Mathematik, Physik, Philosophie, Pädagogik und Französisch. Nach jahrzehntelanger Arbeit in der Datenverarbeitung in Sindelfingen, Böblingen und Stuttgart nahm er nach seiner Pensionierung 1996 an der Uni Tübingen ein Zweitstudium der Allgemeinen und Vergleichenden Sprachwissenschaft und der Islamkunde auf, das er 2001 mit dem Magister Artium und 2003 mit der Promotion abschloss.

Klappentext

This book is a research monograph in the relatively new field of nonparametric regression. It serves as an introduction to the field for graduate students, researchers and statistical consultants in statistics and biostatistics, but is also intended as an overview over some recent research developments in the fixed design case. Basic ideas are developed for various nonparametric curve estimators. The emphasis is on kernel estimators as a unifying concept and on the interplay between theory and practical application. Problems of practical application are illustrated in several examples of analyses of longitudinal medical data sets. These demonstrate the need of including nonparametric regression in addition to the classical parametric regression into the repertoire of practicing statisticians/biostatisticians. One goal of the book is to stimulate the reader to experiment with the methods and to gain experience by applying them. This is facilitated by an Appendix containing several relevant FORTRAN programs.

Inhalt

1. Introduction.- 2. Longitudinal data and regression models.- 2.1 Longitudinal data.- 2.2 Regression models.- 2.3 Longitudinal growth curves.- 3. Nonparametric regression methods.- 3.1 Kernel estimates.- 3.2 Weighted local least squares estimates.- 3.3 Smoothing splines.- 3.4 Orthogonal series estimates.- 3.5 Discussion.- 3.6 Heart pacemaker study.- 4. Kernel and weighted local least squares methods.- 4.1 Mean Squared Error of kernel estimates for curves and derivatives.- 4.2 Asymptotic normality.- 4.3 Boundary effects and Integrated Mean Squared Error.- 4.4 Muscular activity as a function of force.- 4.5 Finite sample comparisons.- 4.6 Equivalence of weighted local regression and kernel estimators.- 5. Optimization of kernel and weighted local regression methods.- 5.1 Optimal designs.- 5.2 Choice of kernel functions.- 5.3 Minimum variance kernels.- 5.4 Optimal kernels.- 5.5 Finite evaluation of higher order kernels.- 5.6 Further criteria for kernels.- 5.7 A hierarchy of smooth optimum kernels.- 5.8 Smooth optimum boundary kernels.- 5.9 Choice of the order of kernels for estimating b? functions.- 6. Multivariate kernel estimators.- 6.1 Definiton and MSE/IMSE.- 6.2 Boundary effects and dimension problem.- 6.3 Rectangular designs and product kernels.- 7. Choice of global and local bandwidths.- 7.1 Overview.- 7.2 Pilot methods.- 7.3 Cross-validation and related methods.- 7.4 Bandwidth choice for derivatives.- 7.5 Confidence intervals for anthropokinetic data.- 7.6 Local versus global bandwidth choice.- 7.7 Weak convergence of a local bandwidth process.- 7.8 Practical local bandwidth choice.- 8. Longitudinal parameters.- 8.1 Comparison of samples of curves.- 8.2 Definition of longitudinal parameters and consistency.- 8.3 Limit distributions.- 9. Nonparametric estimationof the human height growth curve.- 9.1 Introduction.- 9.2 Choice of kernels and bandwidths.- 9.3 Comparison of parametric and nonparametric regression.- 9.4 Estimation of growth velocity and acceleration.- 9.5 Longitudinal parameters for growth curves.- 9.6 Growth spurts.- 10. Further applications.- 10.1 Monitoring and prognosis based on longitudinal medical data.- 10.2 Estimation of heteroscedasticity and prediction intervals.- 10.3 Further developments.- 11. Consistency properties of moving weighted averages.- 11.1 Local weak consistency.- 11.2 Uniform consistency.- 12. FORTRAN routines for kernel smoothing and differentiation.- 12.1 Structure of main routines KESMO and KERN.- 12.2 Listing of programs.- References.