Smooth Tests of Goodness of Fit : Using R

by Rayner, J. C. W.; Thas, O.; Best, D. J.

• ISBN13: 9780470824429
• ISBN10: 0470824425
• Edition: 2nd
• Format: Hardcover
• Copyright: 2009-06-09
• Publisher: Wiley

Summary

Interest in smooth tests has grown as a result of the publication of the first edition of this book and sparked interest in, for example, data-determined order of the tests, the diagnostic properties of the components of the test statistic and model selection techniques.Very practical, with numerous examples, this fully revised and updated new edition: Outlines new research, introduce new methodology and expand the contents to cover virtually any statistical distribution. Provides powerful techniques for assessment of statistical and probabilistic models, with a guide to acceptable alternatives Focuses on a class of procedures demonstrated to work well, presented in a manner that allows the development of assessment tools for new distributions and complex probabilistic models Includes online examples in RReviews of the first edition:"This book gives a very readable account of the smooth tests of goodness of fit. The book can be read by scientists having only an introductory knowledge of statistics. It contains a fairly extensive list of references; research will find it helpful for the further development of smooth tests." --T.K. Chandra, Zentralblatt f8r Mathematik und ihre Grenzgebiete, Band 73, 1/92'"An excellent job of showing how smooth tests (a class of goodness of fit tests) are generally and easily applicable in assessing the validity of models involving statistical distributions....Highly recommended for undergraduate and graduate libraries." --Choice"The book can be read by scientists having only an introductory knowledge of statistics. It contains a fairly extensive list of references; researchers will find it helpful for the further development of smooth tests."--Mathematical Reviews"Very rich in examples . . . Should find its way to the desks of many statisticians." --TechnometricsEssential reading for Researchers and postgraduates carrying out research on goodness-of-fit, statistical and probabilistic model assessment and hypothesis testing.

Table of Contents

Preface	p. xiii
Introduction	p. 1
The Problem Defined	p. 1
A Brief History of Smooth Tests	p. 4
Monograph Outline	p. 9
Examples	p. 10
Pearson's X2 Test	p. 17
Introduction	p. 17
Foundations	p. 17
The Pearson X2 Test - an Update	p. 19
Notation, Definition of the Test, and Class Construction	p. 19
Power Related Properties	p. 21
The Sample Space Partition Approach	p. 24
X2 Tests of Composite Hypotheses	p. 26
Examples	p. 27
Asymptotically Optimal Tests	p. 33
Introduction	p. 33
The Likelihood Ratio, Wald, and Score Tests for a Simple Null Hypothesis	p. 34
The Likelihood Ratio, Wald and Score Tests for Composite Null Hypotheses	p. 38
Generalized Score Tests	p. 47
Neyman Smooth Tests for Simple Null Hypotheses	p. 53
Neyman's ¿2 test	p. 53
Neyman Smooth Tests for Uncategorized Simple Null Hypotheses	p. 55
The Choice of Order	p. 59
Examples	p. 61
EDF Tests	p. 63
Categorized Simple Null Hypotheses	p. 65
Smooth Tests for Completely Specified Multinomials	p. 65
X2 Effective Order	p. 69
Components of X2p	p. 71
Construction of the Components	p. 71
Power Study	p. 72
Diagnostic Tests	p. 75
Cressie and Read Tests	p. 75
Examples	p. 76
Class Construction	p. 81
The Alternatives	p. 82
Results of the Simulation Study	p. 85
Discussion	p. 88
A More Comprehensive Class of Tests	p. 89
Overlapping Cells Tests	p. 91
Neyman Smooth Tests for Uncategorized Composite Null Hypotheses	p. 95
Neyman Smooth Tests for Uncategorized Composite Null Hypotheses	p. 95
Smooth Tests for the Univariate Normal Distribution	p. 102
The Construction of the Smooth Test	p. 102
Simulation Study	p. 103
Examples	p. 105
Relationship with a Test of Thomas and Pierce	p. 109
Smooth Tests for the Exponential Distribution	p. 109
Smooth Tests for Multivariate Normal Distribution	p. 122
Smooth Tests for the Bivariate Poisson Distribution	p. 122
Definitions	p. 122
Score Tests for the Bivariate Poisson Model	p. 123
A Smooth Covariance Test	p. 126
Variance Tests	p. 127
A Competitor for the Index of Dispersion Test	p. 128
Revised Index of Dispersion and Crockett Tests	p. 130
Components of the Rao-Robson X2 Statistic	p. 134
Neyman Smooth Tests for Categorized Composite Null Hypotheses	p. 137
Neyman Smooth Tests for Composite Multinomials	p. 137
Components of the Pearson-Fisher Statistic	p. 142
Composite Overlapping Cells and Cell Focusing X2 Tests	p. 144
A Comparison between the Pearson-Fisher and Rao-Robson X2 Tests	p. 147
Neyman Smooth Tests for Uncategorized Composite Null Hypotheses: Discrete Distributions	p. 151
Neyman Smooth Tests for Discrete Uncategorized Composite Null Hypotheses	p. 151
Smooth and EDF Tests for the Univariate Poisson Distribution	p. 155
Definitions	p. 155
Size and Power Study	p. 157
Examples	p. 160
Smooth and EDF Tests for the Binomial Distribution	p. 163
Definitions	p. 163
Size and Power Study	p. 165
Examples	p. 169
Smooth Tests for the Geometric Distribution	p. 170
Definitions	p. 170
Size and Power Study	p. 171
Examples	p. 175
Construction of Generalized Smooth Tests: Theoretical Contributions	p. 179
Introduction	p. 179
Smooth Test Statistics with Informative Decompositions	p. 180
Sufficient Condition for 'Convenient' Test Statistics	p. 180
Testing for an Exponential Family of Distributions	p. 181
Testing for Distributions not from an Exponential Family	p. 183
Generalized Smooth Tests with Informative Decompositions	p. 183
Uncategorized Distributions	p. 183
Categorized Distributions	p. 186
A Note on the Efficient Score Test	p. 187
Efficiency	p. 187
Diagnostic Component Tests	p. 189
Are Smooth Tests and Their Components Diagnostic?	p. 189
Properly Rescaled Tests	p. 190
Rescaling Outside Exponential Families	p. 191
A Simulation Study	p. 193
Smooth Modelling	p. 199
Introduction	p. 199
Model Selection through Hypothesis Testing	p. 201
Forward Selection and Backward Elimination	p. 201
Smooth Tests for Improved Models	p. 202
Examples	p. 203
Model Selection Based on Loss Functions	p. 206
Loss Functions and Expected Loss	p. 206
AIC and BIC	p. 208
Goodness of Fit Testing after Model Selection	p. 211
Motivation	p. 211
Theory	p. 212
Examples	p. 214
A Final Note	p. 218
Correcting the Barton Density	p. 218
Generalized Smooth Tests for Uncategorized Composite Null Hypotheses	p. 221
Introduction	p. 221
Generalized Smooth Tests for the Logistic Distribution	p. 224
Generalized Smooth Tests for the Laplace Distribution	p. 226
Generalized Smooth Tests for the Extreme Value Distribution	p. 229
Generalized Smooth Tests for the Negative Binomial Distribution	p. 232
Generalized Smooth Tests for the Zero-Inflated Poisson Distribution	p. 234
Generalized Smooth Tests for the Generalized Pareto Distribution	p. 238
Orthonormal Polynomials and Recurrence Relations	p. 243
Parametric Bootstrap p-Values	p. 247
Some Details for Particular Distributions	p. 249
The One-Parameter Logistic Distribution	p. 249
The Orthonormal Polynomials	p. 249
Estimation of the Nuisance Parameters	p. 249
Asymptotic Covariance Matrix of (&Vhat;1,...,&Vhat;4) with ML	p. 249
Asymptotic Covariance Matrix of (&Vtilde;2,...,&Vtilde;4) with MOM	p. 250
The Two-Parameter Logistic Distribution	p. 250
The Orthonormal Polynomials	p. 250
Estimation of the Nuisance Parameters	p. 250
Asymptotic Covariance Matrix of (&Vhat;1,...,&Vhat;4) with ML	p. 251
Asymptotic Covariance Matrix of (&Vtilde;3,...,&Vtilde;4) with MOM	p. 251
The Zero-Inflated Poisson Distribution	p. 251
The Orthonormal Polynomials	p. 251
Estimation of the Nuisance Parameters	p. 252
Asymptotic Covariance Matrix of (&Vtilde;3,...,&Vtilde;4) with MOM	p. 252
The Laplace Distribution	p. 253
The Orthonormal Polynomials	p. 253
Estimation of the Nuisance Parameters	p. 253
Asymptotic Covariance Matrix of (&Vhat;1,...,&Vhat;4) with ML	p. 253
Asymptotic Covariance Matrix of (&Vtilde;3,...,&Vtilde;4) with MOM	p. 254
The Extreme Value Distribution	p. 254
The Orthonormal Polynomials	p. 254
Estimation of the Nuisance Parameters	p. 254
Asymptotic Covariance Matrix of (&Vhat;1,...,&Vhat;4) with ML	p. 255
Asymptotic Covariance Matrix of (&Vtilde;3,...,&Vtilde;4) with MOM	p. 255
The Negative Binomial Distribution	p. 255
The Orthonormal Polynomials	p. 255
Estimation of the Nuisance Parameters	p. 256
Asymptotic Covariance Matrix of (&Vtilde;1,...,&Vtilde;4) with ML	p. 255
Asymptotic Covariance Matrix of (&Vtilde;3, &Vtilde;4) with MOM	p. 255
The Generalized Pareto Distribution	p. 256
The Orthonormal Polynomials	p. 256
Estimation of the Nuisance Parameters	p. 257
Asymptotic Covariance Matrix of (&Vtilde;3,...,&Vtilde;4) with MOM	p. 257
References	p. 259
Subject Index	p. 269
Author Index	p. 271
Example Index	p. 273
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