This new edition continues to serve as a comprehensive guide to modern and classical methods of statistical computing. The book is comprised of four main parts spanning the field:
- Optimization
- Integration and Simulation
- Bootstrapping
- Density Estimation and Smoothing
Within these sections,each chapter includes a comprehensive introduction and step-by-step implementation summaries to accompany the explanations of key methods. The new edition includes updated coverage and existing topics as well as new topics such as adaptive MCMC and bootstrapping for correlated data. The book website now includes comprehensive R code for the entire book. There are extensive exercises, real examples, and helpful insights about how to use the methods in practice.
Table of Contents:
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
