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 des matières
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. omega2, 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. Mc Nemar’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
A propos de l’auteur
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.
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