Nonparametric Tests for Complete Data

Statistical analysis of data sets usually involves construction of a statistical model of the distribution of data within the available sample – and by extension the distribution of all data of the same category in the world. Statistical models are either parametric or non-parametric – this distinction is based on whether or not the model can be described in terms of a finite-dimensional parameter – and the models must be tested to ascertain whether or not they conform to the data, or are accurate.

This book addresses the testing of hypotheses in non-parametric models in the general case for complete data samples. Classical non-parametric tests (goodness-of-fit, homogeneity, randomness, independence) of complete data are considered, and explained. Tests featured include the chi-squared and modified chi-squared tests, rank and homogeneity tests, and most of the test results are proved, with real applications illustrated using examples. The incorrect use of many tests, and their application using commonly deployed statistical software is highlighted and discussed.

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

1.3. 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

2.5. 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

3.6. Bibliographic notes 104

3.7. Exercises106

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

4.4. Randomness tests 139

4.5. Rank homogeneity tests for two independent samples 146

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

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

5.2. Runs test 221

5.3. McNemar’s test 231

5.4. Cochran test 238

5.5. Special goodness-of-fit tests 245

5.6. Bibliographic notes 268

5.7. Exercises 269

5.8. Answers 271

APPENDICES 275

Appendix A. Parametric Maximum Likelihood 277

Appendix B. Notions from the Theory of 281

BBibliography 293

Index 305

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.