ISBN-10:
1848212690
ISBN-13:
9781848212695
Pub. Date:
11/01/2010
Publisher:
Wiley
Nonparametric Tests for Complete Data / Edition 1

Nonparametric Tests for Complete Data / Edition 1

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Product Details

ISBN-13: 9781848212695
Publisher: Wiley
Publication date: 11/01/2010
Series: ISTE Series , #539
Pages: 320
Product dimensions: 6.20(w) x 9.20(h) x 1.00(d)

About the Author

Vilijandas Bagdonavicius is Professor of Mathematics at the University of Vilnius in Lithuania. His main research areas are statistics, reliability and survival analysis.

Julius Kruopis is Associate Professor of Mathematics at the University of Vilnius in Lithuania. His main research areas are statistics and quality control.

Mikhail S. Nikulin is a member of the Institute of Mathematics in Bordeaux, France.

Table of Contents

Preface xi

Terms and Notation xv

Chapter 1 Introduction 1

1.1 Statistical hypotheses 1

1.2 Examples of hypotheses in non-parametric models 2

1.2.1 Hypotheses on the probability distribution of data elements 2

1.2.2 Independence hypotheses 4

1.2.3 Randomness hypothesis 4

1.2.4 Homogeneity hypotheses 4

1.2.5 Median value hypotheses 5

1.3 Statistical tests 5

1.4 P-value 7

1.5 Continuity correction 10

1.6 Asymptotic relative efficiency 13

Chapter 2 Chi-squared Tests 17

2.1 Introduction 17

2.2 Pearson's goodness-of-fit test: simple hypothesis 17

2.3 Pearson's goodness-of-fit test: composite hypothesis 26

2.4 Modified chi-squared test for composite hypotheses 34

2.4.1 General case 35

2.4.2 Goodness-of-fit for exponential distributions 41

2.4.3 Goodness-of-fit for location-scale and shape-scale families 43

2.5 Chi-squared test for independence 52

2.6 Chi-squared test for homogeneity 57

2.7 Bibliographic notes 64

2.8 Exercises 64

2.9 Answers 72

Chapter 3 Goodness-of-fit Tests Based on Empirical Processes 77

3.1 Test statistics based on the empirical process 77

3.2 Kolmogorov-Smirnov test 82

3.3 ω2, Cramér-von-Mises and Andersen-Darling tests 86

3.4 Modifications of Kolmogorov-Smirnov, Cramér-von-Mises and Andersen-Darling tests: composite hypotheses 91

3.5 Two-sample tests 98

3.5.1 Two-sample Kolmogorov-Smirnov tests 98

3.5.2 Two-sample Cramér-von-Mises test 103

3.6 Bibliographic notes 104

3.7 Exercises 106

3.8 Answers 109

Chapter 4 Rank Tests 111

4.1 Introduction 111

4.2 Ranks and their properties 112

4.3 Rank tests for independence 117

4.3.1 Spearman's independence test 117

4.3.2 Kendall's independence test 124

4.3.3 ARE of Kendall's independence test with respect to Pearson's independence test under normal distribution 133

4.3.4 Normal scores independence test 137

4.4 Randomness tests 139

4.4.1 Kendall's and Spearman's randomness tests 140

4.4.2 Bartels-Von Neuman randomness test 143

4.5 Rank homogeneity tests for two independent samples 146

4.5.1 Wilcoxon (Mann-Whitney-Wilcoxon) rank sum test 146

4.5.2 Power of the Wilcoxon rank sum test against location alternatives 153

4.5.3 ARE of the Wilcoxon rank sum test with respect to the asymptotic Student's test 155

4.5.4 Van der Warden's test 161

4.5.5 Rank homogeneity tests for two independent samples under a scale alternative 163

4.6 Hypothesis on median value: the Wilcoxon signed ranks test 168

4.6.1 Wilcoxon's signed ranks tests 168

4.6.2 ARE of the Wilcoxon signed ranks test with respect to Student's test 177

4.7 Wilcoxon's signed ranks test for homogeneity of two related samples 180

4.8 Test for homogeneity of several independent samples: Kruskal-Wallis test 181

4.9 Homogeneity hypotheses for k related samples: Friedman test 191

4.10 Independence test based on Kendall's concordance coefficient 204

4.11 Bibliographic notes 208

4.12 Exercises 209

4.13 Answers 212

Chapter 5 Other Non-parametric Tests 215

5.1 Sign test 215

5.1.1 Introduction: parametric sign test 215

5.1.2 Hypothesis on the nullity of the medians of the differences of random vector components 218

5.1.3 Hypothesis on the median value 220

5.2 Runs test 221

5.2.1 Runs test for randomness of a sequence of two opposite events 223

5.2.2 Runs test for randomness of a sample 226

5.2.3 Wald-Wolfowitz test for homogeneity of two independent samples 228

5.3 McNemar's test 231

5.4 Cochran test 238

5.5 Special goodness-of-fit tests 245

5.5.1 Normal distribution 245

5.5.2 Exponential distribution 253

5.5.3 Weibull distribution 260

5.5.4 Poisson distribution 262

5.6 Bibliographic notes 268

5.7 Exercises 269

5.8 Answers 271

Appendices 275

Appendix A Parametric Maximum Likelihood Estimators: Complete Samples 277

Appendix B Notions from the Theory of Stochastic Processes 281

B.1 Stochastic process 281

B.2 Examples of stochastic processes 282

B.2.1 Empirical process 282

B.2.2 Gauss process 283

B.2.3 Wiener process (Brownian motion) 283

B.2.4 Brownian bridge 284

B.3 Weak convergence of stochastic processes 285

B.4 Weak invariance of empirical processes 286

B.5 Properties of Brownian motion and Brownian bridge 287

Bibliography 293

Index 305

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