Nonparametric Tests for Complete Data

by Bagdonavicius, Vilijandas; Kruopis, Julius; Nikulin, Mikhail S.

• ISBN13: 9781848212695
• ISBN10: 1848212690
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-01-18
• Publisher: Wiley-ISTE

Summary

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.

Author Biography

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	p. xi
Terms and Notation	p. xv
Introduction	p. 1
Statistical hypotheses	p. 1
Examples of hypotheses in non-parametric models	p. 2
Hypotheses on the probability distribution of data elements	p. 2
Independence hypotheses	p. 4
Randomness hypothesis	p. 4
Homogeneity hypotheses	p. 4
Median value hypotheses	p. 5
Statistical tests	p. 5
P-value	p. 7
Continuity correction	p. 10
Asymptotic relative efficiency	p. 13
Chi-squared Tests	p. 17
Introduction	p. 17
Pearson's goodness-of-fit test: simple hypothesis	p. 17
Pearson's goodness-of-fit test: composite hypothesis	p. 26
Modified chi-squared test for composite hypotheses	p. 34
General case	p. 35
Goodness-of-fit for exponential distributions	p. 41
Goodness-of-fit for location-scale and shape-scale families	p. 43
Chi-squared test for independence	p. 52
Chi-squared test for homogeneity	p. 57
Bibliographic notes	p. 64
Exercises	p. 64
Answers	p. 72
Goodness-of-fit Tests Based on Empirical Processes	p. 77
Test statistics based on the empirical process	p. 77
Kolmogorov-Smirnov test	p. 82
¿2, Cramér-von-Mises and Andersen-Darling tests	p. 86
Modifications of Kolmogorov-Smirnov, Cramér-von-Mises and Andersen-Darling tests: composite hypotheses	p. 91
Two-sample tests	p. 98
Two-sample Kolmogorov-Smirnov tests	p. 98
Two-sample Cramér-von-Mises test	p. 103
Bibliographic notes	p. 104
Exercises	p. 106
Answers	p. 109
Rank Tests	p. 111
Introduction	p. 111
Ranks and their properties	p. 112
Rank tests for independence	p. 117
Spearman's independence test	p. 117
Kendall's independence test	p. 124
ARE of Kendall's independence test with respect to Pearson's independence test under normal distribution	p. 133
Normal scores independence test	p. 137
Randomness tests	p. 139
Kendall's and Spearman's randomness tests	p. 140
Bartels-Von Neuman randomness test	p. 143
Rank homogeneity tests for two independent samples	p. 146
Wilcoxon (Mann-Whitney-Wilcoxon) rank sum test	p. 146
Power of the Wilcoxon rank sum test against location alternatives	p. 153
ARE of the Wilcoxon rank sum test with respect to the asymptotic Student's test	p. 155
Van der Warden's test	p. 161
Rank homogeneity tests for two independent samples under a scale alternative	p. 163
Hypothesis on median value: the Wilcoxon signed ranks test	p. 168
Wilcoxon's signed ranks tests	p. 168
ARE of the Wilcoxon signed ranks test with respect to Student's test	p. 177
Wilcoxon's signed ranks test for homogeneity of two related samples	p. 180
Test for homogeneity of several independent samples: Kruskal-Wallis test	p. 181
Homogeneity hypotheses for k related samples: Friedman test	p. 191
Independence test based on Kendall's concordance coefficient	p. 204
Bibliographic notes	p. 208
Exercises	p. 209
Answers	p. 212
Other Non-parametric Tests	p. 215
Sign test	p. 215
Introduction: parametric sign test	p. 215
Hypothesis on the nullity of the medians of the differences of random vector components	p. 218
Hypothesis on the median value	p. 220
Runs test	p. 221
Runs test for randomness of a sequence of two opposite events	p. 223
Runs test for randomness of a sample	p. 226
Wald-Wolfowitz test for homogeneity of two independent samples	p. 228
McNemar's test	p. 231
Cochran test	p. 238
Special goodness-of-fit tests	p. 245
Normal distribution	p. 245
Exponential distribution	p. 253
Weibull distribution	p. 260
Poisson distribution	p. 262
Bibliographic notes	p. 268
Exercises	p. 269
Answers	p. 271
Appendices	p. 275
Parametric Maximum Likelihood Estimators: Complete Samples	p. 277
Notions from the Theory of Stochastic Processes	p. 281
Stochastic process	p. 281
Examples of stochastic processes	p. 282
Empirical process	p. 282
Gauss process	p. 283
Wiener process (Brownian motion)	p. 283
Brownian bridge	p. 284
Weak convergence of stochastic processes	p. 285
Weak invariance of empirical processes	p. 286
Properties of Brownian motion and Brownian bridge	p. 287
Bibliography	p. 293
Index	p. 305
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