Statistical Tests Of Nonparametric Hypotheses: Asymptotic Theory

by Odile Pons

Overview

An overview of the asymptotic theory of optimal nonparametric tests is presented in this book. It covers a wide range of topics: Neyman-Pearson and LeCam's theories of optimal tests, the theories of empirical processes and kernel estimators with extensions of their applications to the asymptotic behavior of tests for distribution functions, densities and curves of the nonparametric models defining the distributions of point processes and diffusions. With many new test statistics developed for smooth curves, the reliance on kernel estimators with bias corrections and the weak convergence of the estimators are useful to prove the asymptotic properties of the tests, extending the coverage to semiparametric models. They include tests built from continuously observed processes and observations with cumulative intervals.

Product Details

ISBN-13:	9789814531740
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/20/2013
Pages:	304
Product dimensions:	6.20(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface v

1 Introduction 1

1.1 Definitions 1

1.2 Rank tests and empirical distribution functions 7

1.3 Hypotheses of the tests 9

1.4 Weak convergence of the test statistics 12

1.5 Tests for densities and curves 16

1.6 Asymptotic levels of tests 19

1.7 Permutation and bootstrap tests 22

1.8 Relative efficiency of tests 24

2 Asymptotic theory 27

2.1 Parametric tests 27

2.2 Parametric likelihood ratio tests 31

2.3 Likelihood ratio tests against local alternatives 36

2.4 Nonparametric likelihood ratio tests 43

2.5 Nonparametric tests for empirical functionals 48

2.6 Tests of homogeneity 51

2.7 Mixtures of exponential distributions 56

2.8 Nonparametric bootstrap tests 61

2.9 Exercises 62

3 Nonparametric tests for one sample 65

3.1 Introduction 65

3.2 Kolmogorov-Smirnov tests for a distribution function 66

3.3 Tests for symmetry of a density 68

3.3.1 Kolmogorov-Smirnov tests for symmetry 69

3.3.2 Semi-parametric tests, with an unknown center 72

3.3.3 Rank test for symmetry 74

3.4 Tests about the form of a density 77

3.5 Goodness of fit test in biased length models 82

3.6 Goodness of fit tests for a regression function 85

3.7 Tests about the form of a regression function 90

3.8 Tests based on observations by intervals 91

3.8.1 Goodness of fit tests for a density 92

3.8.2 Goodness of fit tests for a regression function 97

3.8.3 Tests of symmetry for a density 101

3.8.4 Tests of a monotone density 102

3.9 Exercises 103

4 Two-sample tests 105

4.1 Introduction 105

4.2 Tests of independence 107

4.2.1 Kolmogorov-Smirnov and Cramer-von Mises tests 107

4.2.2 Tests based on the dependence function 109

4.2.3 Tests based on the conditional distribution 113

4.3 Test of homogeneity 115

4.4 Goodness of fit tests in R² 121

4.5 Tests of symmetry for a bivariate density 123

4.6 Tests about the form of densities 125

4.7 Comparison of two regression curves 131

4.8 Tests based on observations by intervals 134

4.8.1 Test of independence 134

4.8.2 Test of homogeneity 136

4.8.3 Comparison of two regression curves 137

4.9 Exercises 138

5 Multi-dimensional tests 141

5.1 Introduction 141

5.2 Tests of independence 142

5.3 Test of homogeneity of k sub-samples 146

5.4 Test of homogeneity of k rescaled distributions 151

5.5 Test of homogeneity of several variables of R^k 153

5.6 Test of equality of marginal distributions 154

5.7 Test of exchangeable components for a random variable 155

5.8 Tests in single-index models 156

5.9 Comparison of k curves 162

5.10 Tests in proportional odds models 164

5.11 Tests for observations by intervals 166

5.11.1 Test of independence 167

5.11.2 Test of homogeneity 168

5.11.3 Comparison of k regression curves 169

5.12 Competing risks 170

5.13 Tests for Markov renewal processes 171

5.14 Tests in R^k_n as k_n tends to infinity 172

5.15 Exercises 176

6 Nonparametric tests for processes 177

6.1 Introduction 177

6.2 Goodness of fit tests for an ergodic process 178

6.3 Poisson process 180

6.4 Poisson processes with scarce jumps 188

6.5 Point processes in R₊ 190

6.6 Marked point processes 198

6.7 Spatial Poisson processes 200

6.8 Tests of stationarity for point processes 202

6.9 Diffusion processes 204

6.10 Comparison of diffusion processes 213

6.11 Exercises 215

7 Nonparametric tests under censoring or truncation 217

7.1 Introduction 217

7.2 Comparison of right-censored distributions 220

7.3 Likelihood ratio test of homogeneity 224

7.4 Tests of homogeneity against general local alternatives 231

7.5 Goodness of fit for the hazard functions ratio 233

7.6 Tests of comparison of k samples 238

7.7 Goodness of fit tests for k samples 242

7.8 Tests of independence of two censored variables 245

7.9 Comparison of two bivariate distributions 247

7.10 Tests for left-censored samples 249

7.11 Tests for the mean residual life and excess life 251

7.12 Tests for right or left-truncated samples 253

7.13 Comparison of censored or truncated regression curves 254

7.14 Observation in random intervals 256

7.15 Exercises 257

8 Sequential tests 259

8.1 Introduction 259

8.2 Definitions and properties 261

8.3 Sequential likelihood ratio test 265

8.4 Sequential algorithms for test statistics 267

8.5 Properties of the record variables 270

8.6 Sequential tests for point processes 272

8.7 Sequential tests for hazard functions 274

8.8 Sequential tests for regressions and diffusion processes 276

Appendix A Functional estimation and probability 279

A.1 Chernov's theorem 279

A.2 Martingales in R₊ 279

A.3 Weak convergence 281

A.4 Algebra 283

A.5 Estimation of densities and curves 283

Bibliography 287

Index 291