Nonparametric Tests for Complete Data

This book concerns testing hypotheses in non-parametric models. Classical non-parametric tests (goodness-of-fit, homogeneity, randomness, independence) of complete data are considered. Most of the test results are proved and real applications are illustrated using examples. Theories and exercises are provided. The incorrect use of many tests applying most statistical software is highlighted and discussed.

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 1.3. 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