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1997 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Nonparametric Smoothing and Lack-of-Fit Tests

verfasst von: Jeffrey D. Hart

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

The The primary primary aim aim of of this this book book is is to to explore explore the the use use of of nonparametric nonparametric regres­ regres­ sion sion (i. e. , (i. e. , smoothing) smoothing) methodology methodology in in testing testing the the fit fit of of parametric parametric regression regression models. models. It It is is anticipated anticipated that that the the book book will will be be of of interest interest to to an an audience audience of of graduate graduate students, students, researchers researchers and and practitioners practitioners who who study study or or use use smooth­ smooth­ ing ing methodology. methodology. Chapters Chapters 2-4 2-4 serve serve as as a a general general introduction introduction to to smoothing smoothing in in the the case case of of a a single single design design variable. variable. The The emphasis emphasis in in these these chapters chapters is is on on estimation estimation of of regression regression curves, curves, with with hardly hardly any any mention mention of of the the lack-of­ lack-of­ fit fit problem. problem. As As such, such, Chapters Chapters 2-4 2-4 could could be be used used as as the the foundation foundation of of a a graduate graduate level level statistics statistics course course on on nonparametric nonparametric regression. regression.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
The estimation of functions is a pervasive statistical problem in scientific endeavors. This book provides an introduction to some nonparametric methods of function estimation, and shows how they can be used to test the adequacy of parametric function estimates. The settings in which function estimation has been studied are many, and include probability density estimation, time series spectrum estimation, and estimation of regression functions. The present treatment will deal primarily with regression, but many of the ideas and methods to be discussed have applications in other areas as well.
Jeffrey D. Hart
2. Some Basic Ideas of Smoothing
Abstract
In its broadest sense, smoothing is the very essence of statistics. To smooth is to sand away the rough edges from a set of data. More precisely, the aim of smoothing is to remove data variability that has no assignable cause and to thereby make systematic features of the data more apparent. In recent years the term smoothing has taken on a somewhat more specialized meaning in the statistical literature. Smoothing has become synonomous with a variety of nonparametric methods used in the estimation of functions, and it is in this sense that we shall use the term. Of course, a primary aim of smoothing in this latter sense is still to reveal interesting data features. Some major accounts of smoothing methods in various contexts may be found in Priestley (1981), Devroye and Györfi (1985), Silverman (1986), Eubank (1988), Härdle (1990), Wahba (1990), Scott (1992), Tarter and Lock (1993), Green and Silverman (1994), Wand and Jones (1995) and Fan and Gijbels (1996).
Jeffrey D. Hart
3. Statistical Properties of Smoothers
Abstract
The smoothers discussed in Chapter 2 provide very useful descriptions of regression data. However, when we use smoothers to formally estimate a regression function, it becomes important to understand their statistical properties. In this chapter we discuss issues such as mean squared error and sampling distribution of an estimator, and using smoothers to obtain confidence intervals for values of the regression function. We will consider two types of smoothers: Gasser-Müller type kernel estimators and tapered Fourier series. We choose to focus on Gasser-Müller rather than NadarayaWatson type kernel smoothers since the latter have a more complicated bias representation. Among Fourier series estimators the emphasis will be on truncated series estimators, since there are certain practical and theoretical advantages to using an estimator with a discrete smoothing parameter.
Jeffrey D. Hart
4. Data-Driven Choice of Smoothing Parameters
Abstract
This chapter is devoted to the problem of choosing the smoothing parameter of a nonparametric regression estimator, a problem that plays a major role in the remainder of this book. We will use S to denote a generic smoothing parameter when we are not referring to a particular type of smoother. The sophistication of the technique used to choose S will depend on the data analyst’s reasons for fitting a nonparametric smooth to the data. If one wishes a smooth to be merely a descriptive device, then the “by eye” technique may be satisfactory. Here, one looks at several smooths corresponding to different values of S and chooses one (or more) which display interesting features of the data. In doing so, the data analyst is not necessarily saying that these features are verification of similar ones in a population curve; he only wishes to describe aspects of the data that may warrant further investigation.
Jeffrey D. Hart
5. Classical Lack-of-Fit Tests
Abstract
We now turn our attention to the problem of testing the fit of a parametric regression model. Ultimately, our purpose is to show how the nonparametric smoothing methods encountered in the previous three chapters can be useful in this regard. We begin, however, by considering some classical methods for checking model fit. This is done to provide some historical perspective and also to facilitate comparisons between smoothing-based and classical methods.
Jeffrey D. Hart
6. Lack-of-Fit Tests Based on Linear Smoothers
Abstract
We are now in a position to begin our study of lack-of fit tests based on smoothing methodology.
Jeffrey D. Hart
7. Testing for Association via Automated Order Selection
Abstract
The tests in Chapter 6 assumed a fixed smoothing parameter. In Chapters 7 and 8 we will discuss tests based on data-driven smoothing parameters. The current chapter deals with testing the “no-effect” hypothesis, and Chapter 8 treats more general parametric hypotheses. The methodology proposed in Chapter 7 makes use of an orthogonal series representation for r. In principle any series representation could be used, but for now we consider only trigonometric series. This is done for the sake of clarity and to make the ideas less abstract. Section 7.8 discusses the use of other types of orthogonal series.
Jeffrey D. Hart
8. Data-Driven Lack-of-Fit Tests for General Parametric Models
Abstract
In this chapter we consider testing the fit of parametric models of a more general nature than the constant mean model of Chapter 7. We begin with the case of a linear model, i.e., the case where r is hypothesized to be a linear combination of known functions. The fit of such models can be tested by applying the methods of Chapter 7 to residuals. It will be argued that test statistics generally have the same distributions they had in Chapter 7 if least squares is used to estimate model parameters.
Jeffrey D. Hart
9. Extending the Scope of Application
Abstract
To this point we have dealt with a somewhat limited setting, namely fixeddesign regression with but a single x-variable. In Chapter 9 it will be shown that the tests discussed in Chapters 7 and 8 have a much wider range of application. We shall consider some of the settings in which order selection tests are potentially useful. Our treatment of each setting will be more speculative and not nearly so comprehensive as in Chapters 7 and 8. The goal is not to state and prove an array of theorems on asymptotic distribution theory, but rather to provide a sense of the scope of application of order selection tests.
Jeffrey D. Hart
10. Some Examples
Abstract
In this final chapter we make use of order selection tests in analyzing some actual sets of data. In Section 10.2 tests of linearity are performed on the Babinet data, which were encountered in Section 6.4.2. We also consider the problem of selecting a good model for these data, and perform a test of homoscedasticity. In Section 10.3 order selection tests are used in an analysis of hormone level spectra. Section 10.4 shows how the order selection test can enhance the scatter plots corresponding to a set of multivariate data. Finally, in Section 10.5, the order selection test is used to test whether a multiple regression model has an additive structure.
Jeffrey D. Hart
Backmatter
Metadaten
Titel
Nonparametric Smoothing and Lack-of-Fit Tests
verfasst von
Jeffrey D. Hart
Copyright-Jahr
1997
Verlag
Springer New York
Electronic ISBN
978-1-4757-2722-7
Print ISBN
978-1-4757-2724-1
DOI
https://doi.org/10.1007/978-1-4757-2722-7