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Contents
Preface xiii
1 Introduction 1
1.1 Effciency of Nonparametric Methods 3
1.2 Overconfdence Bias 5
1.3 Computing with MATLAB 5
1.4 Exercises 7
References 7
2 Probability Basics 9
2.1 Helpful Functions 9
2.2 Events, Probabilities and Random Variables 11
2.3 Numerical Characteristics of Random Variables 12
2.4 Discrete Distributions 14
2.5 Continuous Distributions 17
2.6 Mixture Distributions 23
2.7 Exponential Family of Distributions 25
2.8 Stochastic Inequalities 26
2.9 Convergence of Random Variables 28
vii
viii CONTENTS
2.10 Exercises 31
References 32
3 Statistics Basics 33
3.1 Estimation 33
3.2 Empirical Distribution Function 34
3.3 Statistical Tests 36
3.4 Exercises 45
References 46
4 Bayesian Statistics 47
4.1 The Bayesian Paradigm 47
4.2 Ingredients for Bayesian Inference 48
4.3 Bayesian Computation and Use of WinBUGS 61
4.4 Exercises 63
References 67
5 Order Statistics 69
5.1 Joint Distributions of Order Statistics 70
5.2 Sample Quantiles 72
5.3 Tolerance Intervals 73
5.4 Asymptotic Distributions of Order Statistics 75
5.5 Extreme Value Theory 76
5.6 Ranked Set Sampling 76
5.7 Exercises 77
References 80
6 Goodness of Fit 81
6.1 Kolmogorov-Smirnov Test Statistic 82
6.2 Smirnov Test to Compare Two Distributions 86
6.3 Specialized Tests 89
6.4 Probability Plotting 97
6.5 Runs Test 100
6.6 Meta Analysis 106
6.7 Exercises 109
CONTENTS ix
References 113
7 Rank Tests 115
7.1 Properties of Ranks 117
7.2 Sign Test 118
7.3 Spearman Coe±cient of Rank Correlation 122
7.4 Wilcoxon Signed Rank Test 126
7.5 Wilcoxon (Two-Sample) Sum Rank Test 129
7.6 Mann-Whitney U Test 131
7.7 Test of Variances 133
7.8 Exercises 135
References 139
8 Designed Experiments 141
8.1 Kruskal-Wallis Test 141
8.2 Friedman Test 145
8.3 Variance Test for Several Populations 148
8.4 Exercises 149
References 152
9 Categorical Data 153
9.1 Chi-Square and Goodness-of-Fit 155
9.2 Contingency Tables 159
9.3 Fisher Exact Test 163
9.4 MCNemar Test 164
9.5 Cochran's Test 167
9.6 Mantel-Haenszel Test 167
9.7 CLT for Multinomial Probabilities 171
9.8 Simpson's Paradox 172
9.9 Exercises 173
References 180
10 Estimating Distribution Functions 183
10.1 Introduction 183
10.2 Nonparametric Maximum Likelihood 184
x CONTENTS
10.3 Kaplan-Meier Estimator 185
10.4 Con¯dence Interval for F 192
10.5 Plug-in Principle 193
10.6 Semi-Parametric Inference 195
10.7 Empirical Processes 197
10.8 Empirical Likelihood 198
10.9 Exercises 201
References 203
11 Density Estimation 205
11.1 Histogram 206
11.2 Kernel and Bandwidth 207
11.3 Exercises 213
References 215
12 Beyond Linear Regression 217
12.1 Least Squares Regression 218
12.2 Rank Regression 219
12.3 Robust Regression 221
12.4 Isotonic Regression 227
12.5 Generalized Linear Models 230
12.6 Exercises 237
References 240
13 Curve Fitting Techniques 241
13.1 Kernel Estimators 243
13.2 Nearest Neighbor Methods 247
13.3 Variance Estimation 249
13.4 Splines 251
13.5 Summary 257
13.6 Exercises 258
References 260
14 Wavelets 263
14.1 Introduction to Wavelets 263
CONTENTS xi
14.2 How Do the Wavelets Work? 266
14.3 Wavelet Shrinkage 273
14.4 Exercises 281
References 283
15 Bootstrap 285
15.1 Bootstrap Sampling 285
15.2 Nonparametric Bootstrap 287
15.3 Bias Correction for Nonparametric Intervals 292
15.4 The Jackknife 295
15.5 Bayesian Bootstrap 296
15.6 Permutation Tests 298
15.7 More on the Bootstrap 302
15.8 Exercises 302
References 304
16 EM Algorithm 307
16.1 Fisher's Example 309
16.2 Mixtures 311
16.3 EM and Order Statistics 315
16.4 MAP via EM 317
16.5 Infection Pattern Estimation 318
16.6 Exercises 319
References 321
17 Statistical Learning 323
17.1 Discriminant Analysis 324
17.2 Linear Classi¯cation Models 326
17.3 Nearest Neighbor Classi¯cation 329
17.4 Neural Networks 333
17.5 Binary Classi¯cation Trees 338
17.6 Exercises 346
References 346
18 Nonparametric Bayes 349
xii CONTENTS
18.1 Dirichlet Processes 350
18.2 Bayesian Categorical Models 357
18.3 In¯nitely Dimensional Problems 360
18.4 Exercises 364
References 366
A MATLAB 369
A.1 Using MATLAB 369
A.2 Matrix Operations 372
A.3 Creating Functions in MATLAB 374
A.4 Importing and Exporting Data 375
A.5 Data Visualization 380
A.6 Statistics 386
B WinBUGS 397
B.1 Using WinBUGS 398
B.2 Built-in Functions 401
MATLAB Index 405
Author Index 409
Subject Index 413