Computational Statistics

A valuable new edition of the complete guide to modernstatistical computing

Computational Statistics, Second Edition continues toserve as a comprehensive guide to the theory and practice ofstatistical computing. Like its predecessor, the new edition spansa broad range of modern and classic topics including optimization,integration, Monte Carlo methods, bootstrapping, density estimationand smoothing. Algorithms are explained both conceptually and byusing step–by–step descriptions, and are illustrated with detailedexamples and exercises.

Important features of this Second Edition include:

Examples based on real–world applications from various fieldsincluding genetics, ecology, economics, network systems, biology,and medicine Explanations of how computational methods are importantcomponents of major statistical approaches such as Bayesian models,linear and generalized linear models, random effects models,survival models, and hidden Markov models Expanded coverage of Markov chain Monte Carlo methods New topics such as sequential sampling methods, particlefilters, derivative free optimization, bootstrapping dependentdata, and adaptive MCMC New exercises and examples that help readers develop the skillsneeded to apply computational methods to a broad array ofstatistical problems A companion website offering datasets and code in the Rsoftware package

Computational Statistics, Second Edition is perfect foradvanced undergraduate or graduate courses in statistical computingand as a reference for practicing statisticians.

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 EQUATIONS21

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 INTEGRATION151

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

GEOF H. GIVENS, PhD, is Associate Professor in theDepartment of Statistics at Colorado State University. He serves asAssociate Editor for Computational Statistics and DataAnalysis. His research interests include statistical problemsin wildlife conservation biology including ecology, populationmodeling and management, and automated computer facerecognition.

JENNIFER A. HOETING, PhD, is Professor in the Departmentof Statistics at Colorado State University. She is an award–winningteacher who co–leads large research efforts for the NationalScience Foundation. She has served as associate editor for theJournal of the American Statistical Association andEnvironmetrics. Her research interests include spatialstatistics, Bayesian methods, and model selection.

Givens and Hoeting have taught graduate courses on computationalstatistics for nearly twenty years, and short courses to leadingstatisticians and scientists around the world.