Nonparametric Statistics with Applications to Science and Engineering with R

Introduction to the methods and techniques of traditional and modern nonparametric statistics, incorporating R code

Nonparametric Statistics with Applications to Science and Engineering with R presents modern nonparametric statistics from a practical point of view, with the newly revised edition including custom R functions implementing nonparametric methods to explain how to compute them and make them more comprehensible.

Relevant built-in functions and packages on CRAN are also provided with a sample code. R codes in the new edition not only enable readers to perform nonparametric analysis easily, but also to visualize and explore data using R’s powerful graphic systems, such as ggplot2 package and R base graphic system.

The new edition includes useful tables at the end of each chapter that help the reader find data sets, files, functions, and packages that are used and relevant to the respective chapter. New examples and exercises that enable readers to gain a deeper insight into nonparametric statistics and increase their comprehension are also included.

Some of the sample topics discussed in Nonparametric Statistics with Applications to Science and Engineering with R include:

• Basics of probability, statistics, Bayesian statistics, order statistics, Kolmogorov–Smirnov test statistics, rank tests, and designed experiments
• Categorical data, estimating distribution functions, density estimation, least squares regression, curve fitting techniques, wavelets, and bootstrap sampling
• EM algorithms, statistical learning, nonparametric Bayes, WinBUGS, properties of ranks, and Spearman coefficient of rank correlation
• Chi-square and goodness-of-fit, contingency tables, Fisher exact test, MC Nemar test, Cochran’s test, Mantel–Haenszel test, and Empirical Likelihood

Nonparametric Statistics with Applications to Science and Engineering with R is a highly valuable resource for graduate students in engineering and the physical and mathematical sciences, as well as researchers who need a more comprehensive, but succinct understanding of modern nonparametric statistical methods.

Brani Vidakovic is a Professor in the Department of Biomedical Engineering at Georgia Institute of Technology, USA. He received his Ph.D. from Purdue University, USA. Paul Kvam is a Professor in the Department of Mathematics, Richmond University, USA. He is the Associate editor for the American Statistician and the Journal of Quality Technology. Seong-joon Kim is a Manager at Software Center, Doosan Heavy Industries & Construction.

Preface xi

1 Introduction 1

1.1 Efficiency of Nonparametric Methods 2

1.2 Overconfidence Bias 4

1.3 Computing with R 5

1.4 Exercises 6

References 7

2 Probability Basics 9

2.1 Helpful Functions 10

2.2 Events, Probabilities and Random Variables 12

2.3 Numerical Characteristics of Random Variables 13

2.4 Discrete Distributions 14

2.5 Continuous Distributions 18

2.6 Mixture Distributions 24

2.7 Exponential Family of Distributions 26

2.8 Stochastic Inequalities 26

2.9 Convergence of Random Variables 28

2.10 Exercises 32

References 34

3 Statistics Basics 35

3.1 Estimation 36

3.2 Empirical Distribution Function 36

3.3 Statistical Tests 38

3.4 Confidence Intervals 41

3.5 Likelihood 45

3.6 Exercises 49

References 51

4 Bayesian Statistics 53

4.1 The Bayesian Paradigm 53

4.2 Ingredients for Bayesian Inference 54

4.3 Point Estimation 58

4.4 Interval Estimation: Credible Sets 60

4.5 Bayesian Testing 62

4.6 Bayesian Prediction 65

4.7 Bayesian Computation and Use of WinBUGS 67

4.8 Exercises 69

References 73

5 Order Statistics 75

5.1 Joint Distributions of Order Statistics 77

5.2 Sample Quantiles 79

5.3 Tolerance Intervals 79

5.4 Asymptotic Distributions of Order Statistics 81

5.5 Extreme Value Theory 82

5.6 Ranked Set Sampling 83

5.7 Exercises 84

References 87

6 Goodness of Fit 89

6.1 KolmogorovSmirnov Test Statistic 90

6.2 Smirnov Test to Compare Two Distributions 96

6.3 Specialized Tests 99

6.4 Probability Plotting 106

6.5 Runs Test 112

6.6 Meta Analysis 117

6.7 Exercises 121

References 125

7 Rank Tests 127

7.1 Properties of Ranks 128

7.2 Sign Test 130

7.3 Spearman Coefficient of Rank Correlation 135

7.4 Wilcoxon Signed Rank Test 139

7.5 Wilcoxon (TwoSample) Sum Rank Test 142

7.6 MannWhitney U Test 144

7.7 Test of Variances 146

7.8 Walsh Test for Outliers 147

7.9 Exercises 148

References 153

8 Designed Experiments 155

8.1 KruskalWallis Test 156

8.2 Friedman Test 160

8.3 Variance Test for Several Populations 165

8.4 Exercises 166

References 169

9 Categorical Data 171

9.1 ChiSquare and GoodnessofFit 172

9.2 Contingency Tables 178

9.3 Fisher Exact Test 183

9.4 Mc Nemar Test 184

9.5 Cochran’s Test 186

9.6 MantelHaenszel Test 188

9.7 CLT for Multinomial Probabilities 190

9.8 Simpson’s Paradox 191

9.9 Exercises 193

References 200

10 Estimating Distribution Functions 203

10.1 Introduction 203

10.2 Nonparametric Maximum Likelihood 204

10.3 KaplanMeier Estimator 205

10.4 Confidence Interval for F 213

10.5 Plugin Principle 214

10.6 SemiParametric Inference 215

10.7 Empirical Processes 217

10.8 Empirical Likelihood 218

10.9 Exercises 221

References 223

11 Density Estimation 225

11.1 Histogram 226

11.2 Kernel and Bandwidth 228

11.3 Exercises 235

References 236

12 Beyond Linear Regression 237

12.1 Least Squares Regression 238

12.2 Rank Regression 239

12.3 Robust Regression 243

12.4 Isotonic Regression 249

12.5 Generalized Linear Models 252

12.6 Exercises 259

References 261

13 Curve Fitting Techniques 263

13.1 Kernel Estimators 265

13.2 Nearest Neighbor Methods 269

13.3 Variance Estimation 272

13.4 Splines 273

13.5 Summary 279

13.6 Exercises 279

References 282

14 Wavelets 285

14.1 Introduction to Wavelets 285

14.2 How Do the Wavelets Work? 288

14.3 Wavelet Shrinkage 295

14.4 Exercises 304

References 305

15 Bootstrap 307

15.1 Bootstrap Sampling 307

15.2 Nonparametric Bootstrap 309

15.3 Bias Correction for Nonparametric Intervals 315

15.4 The Jackknife 317

15.5 Bayesian Bootstrap 318

15.6 Permutation Tests 320

15.7 More on the Bootstrap 324

15.8 Exercises 325

References 327

16 EM Algorithm 329

16.1 Fisher’s Example 331

16.2 Mixtures 333

16.3 EM and Order Statistics 338

16.4 MAP via EM 339

16.5 Infection Pattern Estimation 341

16.6 Exercises 342

References 343

17 Statistical Learning 345

17.1 Discriminant Analysis 346

17.2 Linear Classification Models 349

17.3 Nearest Neighbor Classification 353

17.4 Neural Networks 355

17.5 Binary Classification Trees 361

17.6 Exercises 368

References 369

18 Nonparametric Bayes 371

18.1 Dirichlet Processes 372

18.2 Bayesian Categorical Models 380

18.3 Infinitely Dimensional Problems 383

18.4 Exercises 387

References 389

A WinBUGS 392

A.1 Using WinBUGS 393

A.2 Builtin

Functions 396

B R Coding 400

B.1 Programming in R 400

B.2 Basics of R 402

B.3 R Commands 403

B.4 R for Statistics 405

R Index 411

Author Index 414

Subject Index 418

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