An introduction to nonparametric statistics

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Introduction......Page 12
   1.1.1.1 Gaussian Distribution......Page 14
   1.1.1.2 Uniform Distribution......Page 15
   1.1.1.4 Cauchy Distribution......Page 16
  1.1.2 Location and Scale Families......Page 17
  1.1.4 ˜2-distribution......Page 18
  1.2.1 Hypothesis Testing......Page 19
   1.2.1.1 One-Sided Hypothesis Tests......Page 20
   1.2.1.2 Two-Sided Hypothesis Tests......Page 22
   1.2.1.3 P-values......Page 23
   1.2.2.1 P-value Inversion......Page 24
   1.2.2.3 A Problematic Example......Page 25
 1.3 Exercises......Page 26
  2.1.1 Estimation Using Averages......Page 28
 2.2 The Need for Distribution-Free Tests......Page 29
 2.3 One-Sample Median Methods......Page 30
  2.3.1 Estimates of the Population Median......Page 31
  2.3.2 Hypothesis Tests Concerning the Population Median......Page 32
  2.3.3 Con dence Intervals for the Median......Page 37
  2.3.4 Inference for Other Quantiles......Page 40
 2.4 Comparing Tests......Page 41
   2.4.1.1 Power......Page 42
   2.4.1.2 Sample and Effect Sizes......Page 43
  2.4.2 Effciency Calculations......Page 44
  2.4.3 Examples of Power Calculations......Page 46
 2.5 Distribution Function Estimation......Page 47
 2.6 Exercises......Page 48
  3.1.1 Two-Sample Approximately Gaussian Inference on Expectations......Page 52
  3.1.2 Approximately Gaussian Dispersion Inference......Page 53
  3.2.1 Null Distributions of General Rank Statistics......Page 54
  3.2.2 Moments of Rank Statistics......Page 55
 3.3 A First Distribution-Free Test......Page 56
 3.4 The Mann-Whitney-Wilcoxon Test......Page 59
   3.4.1.1 Moments and Approximate Normality......Page 61
  3.4.2 Other Scoring Schemes......Page 63
  3.4.3 Using Data as Scores: the Permutation Test......Page 64
 3.5 Empirical Levels and Powers of Two-Sample Tests......Page 66
 3.6 Adaptation to the Presence of Tied Observations......Page 67
  3.8.1 Efficacy of the Gaussian-Theory Test......Page 68
  3.8.2 Efficacy of the Mann-Whitney-Wilcoxon Test......Page 69
  3.8.4 Power for Mann-Whitney-Wilcoxon Testing......Page 70
 3.9 Testing Equality of Dispersion......Page 71
 3.10 Two-Sample Estimation and Con dence Intervals......Page 73
  3.10.1 Inversion of the Mann-Whitney-Wilcoxon Test......Page 74
 3.11 Tests for Broad Alternatives......Page 75
 3.12 Exercises......Page 77
 4.1 Gaussian-Theory Methods......Page 82
  4.1.1 Contrasts......Page 83
  4.1.2 Multiple Comparisons......Page 84
  4.2.1 Moments of General Rank Sums......Page 86
  4.2.2 Construction of a Chi-Square-Distributed Statistic......Page 87
  4.3.1 Kruskal-Wallis Approximate Critical Values......Page 89
 4.4 Other Scores for Multi-Sample Rank Based Tests......Page 91
 4.5 Multiple Comparisons......Page 93
 4.6 Ordered Alternatives......Page 95
  4.7.1 Power of Tests for Ordered Alternatives......Page 97
  4.7.2 Power of Tests for Unordered Alternatives......Page 98
 4.8 E ciency Calculations......Page 103
  4.8.2 Unordered Alternatives......Page 104
 4.9 Exercises......Page 105
  5.1.2 Multiple Group Comparisons......Page 108
 5.2 Nonparametric Paired Comparisons......Page 109
  5.2.1 Estimating the Population Median Di erence......Page 111
  5.2.2 Con dence Intervals......Page 113
  5.2.3 Signed-Rank Statistic Alternative Distribution......Page 114
  5.3.1 Distribution of Rank Sums......Page 115
 5.4 A Generalization of the Test of Friedman......Page 116
  5.4.1 The Balanced Case......Page 117
  5.4.2 The Unbalanced Case......Page 118
 5.5 Multiple Comparisons and Scoring......Page 120
 5.6 Tests for a Putative Ordering in Two-Way Layouts......Page 121
 5.7 Exercises......Page 123
 6.1 Parametric Approach......Page 126
 6.2 Permutation Inference......Page 127
  6.3.1 Rank Correlation......Page 128
  6.3.2 Kendall's......Page 131
   6.4.1.1 Inversion of the Pearson Correlation......Page 134
   6.4.1.2 Inversion of Kendall's T......Page 135
   6.4.1.3 Inversion of the Spearman Correlation......Page 136
 6.5 Exercises......Page 139
  7.1.2 One-Sample Testing......Page 142
 7.2 Nonparametric Multivariate Estimation......Page 143
 7.3 Nonparametric One-Sample Testing Approaches......Page 145
  7.3.1 More General Permutation Solutions......Page 148
 7.5 Two-Sample Methods......Page 149
   7.5.1.1 Permutation Testing......Page 150
   7.5.1.2 Permutation Distribution Approximations......Page 153
 7.6 Exercises......Page 154
 8.1 Histograms......Page 156
 8.2 Kernel Density Estimates......Page 157
 8.3 Exercises......Page 161
 9.1 Standard Regression Inference......Page 162
 9.2 Kernel and Local Regression Smoothing......Page 163
 9.3 Isotonic Regression......Page 167
 9.4 Splines......Page 168
 9.5 Quantile Regression......Page 170
  9.5.1 Fitting the Quantile Regression Model......Page 172
 9.6 Resistant Regression......Page 175
 9.7 Exercises......Page 178
 10.1 The Bootstrap Idea......Page 180
  10.1.1 The Bootstrap Sampling Scheme......Page 181
  10.2.2 Basic Interval......Page 183
  10.2.3 The Percentile Method......Page 184
  10.2.4 BCa Method......Page 185
  10.2.5 Summary So Far, and More Examples......Page 187
 10.3 Bootstrapping Multivariate Data Sets......Page 188
  10.3.1 Regression Models and the Studentized Bootstrap Method......Page 189
  10.3.2 Fixed......Page 191
  10.4.1 Examples of Biases of the Proper Order......Page 194
   10.4.2.2 Correcting the Bias in Quantile Estimators......Page 195
 10.5 Exercises......Page 198
Appendix A: Analysis Using the SAS System......Page 200
Appendix B: Construction of Heuristic Tables and Figures Using R......Page 210
Bibliography......Page 214
Index......Page 224