Statistical Computing with R Book

Preface xv

1 Introduction 1

1.1 Computational Statistics and Statistical Computing 1

1.2 The R Environment 3

1.3 Getting Started with R 4

1.4 Using the R Online Help System 7

1.5 Functions 8

1.6 Arrays, Data Frames, and Lists 9

1.7 Workspace and Files 15

1.8 Using Scripts 17

1.9 Using Packages 18

1.10 Graphics 19

2 Probability and Statistics Review 21

2.1 Random Variables and Probability 21

2.2 Some Discrete Distributions 25

2.3 Some Continuous Distributions 29

2.4 Multivariate Normal Distribution 33

2.5 Limit Theorems 35

2.6 Statistics 35

2.7 Bayes' Theorem and Bayesian Statistics 40

2.8 Markov Chains 42

3 Methods for Generating Random Variables 47

3.1 Introduction 47

3.2 The Inverse Transform Method 49

3.3 The Acceptance-Rejection Method 55

3.4 Transformation Methods 58

3.5 Sums and Mixtures 61

3.6 Multivariate Distributions 69

3.7 Stochastic Processes 82

Exercises 94

4 Visualization of Multivariate Data 97

4.1 Introduction 97

4.2 Panel Displays 97

4.3 Surface Plots and 3D Scatter Plots 100

4.4 Contour Plots 106

4.5 Other 2D Representations of Data 110

4.6 Other Approaches to Data Visualization 115

Exercises 116

5 Monte Carlo Integration and Variance Reduction 119

5.1 Introduction 119

5.2 Monte Carlo Integration 119

5.3 Variance Reduction 126

5.4 Antithetic Variables 128

5.5 Control Variates 132

5.6 Importance Sampling 139

5.7 Stratified Sampling 144

5.8 Stratified Importance Sampling 147

Exercises 149

R Code 152

6 Monte Carlo Methods in Inference 153

6.1 Introduction 153

6.2 Monte Carlo Methods for Estimation 154

6.3 Monte Carlo Methods for Hypothesis Tests 162

6.4 Application174

Exercises 180

7 Bootstrap and Jackknife 183

7.1 The Bootstrap 183

7.2 The Jackknife 190

7.3 Jackknife-after-Bootstrap 195

7.4 Bootstrap Confidence Intervals 197

7.5 Better Bootstrap Confidence Intervals 203

7.6 Application 207

Exercises 212

8 Permutation Tests 215

8.1 Introduction 215

8.2 Tests for Equal Distributions 219

8.3 Multivariate Tests for Equal Distributions 222

8.4 Application 235

Exercises 242

9 Markov Chain Monte Carlo Methods 245

9.1 Introduction 245

9.2 The Metropolis-Hastings Algorithm 247

9.3 The Gibbs Sampler 263

9.4 Monitoring Convergence 266

9.5 Application 271

Exercises 277

R Code 279

10 Probability Density Estimation 281

10.1 Univariate Density Estimation 281

10.2 Kernel Density Estimation 296

10.3 Bivariate and Multivariate Density Estimation 305

10.4 Other Methods of Density Estimation 314

Exercises 314

R Code 317

11 Numerical Methods in R 319

11.1 Introduction 319

11.2 Root-finding in One Dimension 326

11.3 Numerical Integration 330

11.4 Maximum Likelihood Problems 335

11.5 One-dimensional Optimization 338

11.6 Two-dimensional Optimization 342

11.7 The EM Algorithm 345

11.8 Linear Programming - The Simplex Method 348

11.9 Application 349

Exercises 353

A Notation 355

B Working with Data Frames and Arrays 357

B.1 Resampling and Data Partitioning 357

B.2 Subsetting and Reshaping Data 360

B.3 Data Entry and Data Analysis 364

References 375

Index 395