Computational Statistics

GEOF H. GIVENS, PhD, is Associate Professor in the Department of Statistics at Colorado State University. He serves as Associate Editor for Computational Statistics and Data Analysis. His research interests include statistical problems in wildlife conservation biology including ecology, population modeling and management, and automated computer face recognition. JENNIFER A. HOETING, PhD, is Professor in the Department of Statistics at Colorado State University. She is an award-winning teacher who co-leads large research efforts for the National Science Foundation. She has served as associate editor for the Journal of the American Statistical Association and Environmetrics. Her research interests include spatial statistics, Bayesian methods, and model selection. Givens and Hoeting have taught graduate courses on computational statistics for nearly twenty years, and short courses to leading statisticians and scientists around the world.

PREFACE xv

ACKNOWLEDGMENTS xvii

1 REVIEW 1

1.1 Mathematical Notation 1

1.2 Taylor’s Theorem and Mathematical Limit Theory 2

1.3 Statistical Notation and Probability Distributions 4

1.4 Likelihood Inference 9

1.5 Bayesian Inference 11

1.6 Statistical Limit Theory 13

1.7 Markov Chains 14

1.8 Computing 17

PART I OPTIMIZATION

2 OPTIMIZATION AND SOLVING NONLINEAR EQUATIONS 21

2.1 Univariate Problems 22

2.2 Multivariate Problems 34

Problems 54

3 COMBINATORIAL OPTIMIZATION 59

3.1 Hard Problems and NP-Completeness 59

3.2 Local Search 65

3.3 Simulated Annealing 68

3.4 Genetic Algorithms 75

3.5 Tabu Algorithms 85

Problems 92

4 EM OPTIMIZATION METHODS 97

4.1 Missing Data, Marginalization, and Notation 97

4.2 The EM Algorithm 98

4.3 EM Variants 111

Problems 121

PART II INTEGRATION AND SIMULATION

5 NUMERICAL INTEGRATION 129

5.1 Newton–Côtes Quadrature 129

5.2 Romberg Integration 139

5.3 Gaussian Quadrature 142

5.4 Frequently Encountered Problems 146

Problems 148

6 SIMULATION AND MONTE CARLO INTEGRATION 151

6.1 Introduction to the Monte Carlo Method 151

6.2 Exact Simulation 152

6.3 Approximate Simulation 163

6.4 Variance Reduction Techniques 180

Problems 195

7 MARKOV CHAIN MONTE CARLO 201

7.1 Metropolis–Hastings Algorithm 202

7.2 Gibbs Sampling 209

7.3 Implementation 218

Problems 230

8 ADVANCED TOPICS IN MCMC 237

8.1 Adaptive MCMC 237

8.2 Reversible Jump MCMC 250

8.3 Auxiliary Variable Methods 256

8.4 Other Metropolis–Hastings Algorithms 260

8.5 Perfect Sampling 264

8.6 Markov Chain Maximum Likelihood 268

8.7 Example: MCMC for Markov Random Fields 269

Problems 279

PART III BOOTSTRAPPING

9 BOOTSTRAPPING 287

9.1 The Bootstrap Principle 287

9.2 Basic Methods 288

9.3 Bootstrap Inference 292

9.4 Reducing Monte Carlo Error 302

9.5 Bootstrapping Dependent Data 303

9.6 Bootstrap Performance 315

9.7 Other Uses of the Bootstrap 316

9.8 Permutation Tests 317

Problems 319

PART IV DENSITY ESTIMATION AND SMOOTHING

10 NONPARAMETRIC DENSITY ESTIMATION 325

10.1 Measures of Performance 326

10.2 Kernel Density Estimation 327

10.3 Nonkernel Methods 341

10.4 Multivariate Methods 345

Problems 359

11 BIVARIATE SMOOTHING 363

11.1 Predictor–Response Data 363

11.2 Linear Smoothers 365

11.3 Comparison of Linear Smoothers 377

11.4 Nonlinear Smoothers 379

11.5 Confidence Bands 384

11.6 General Bivariate Data 388

Problems 389

12 MULTIVARIATE SMOOTHING 393

12.1 Predictor–Response Data 393

12.2 General Multivariate Data 413

Problems 416

DATA ACKNOWLEDGMENTS 421

REFERENCES 423

INDEX 457