Title Details

SHELEMYAHU ZACKS, PhD, is Distinguished Professor in the Department of Mathematical Sciences at Binghamton University. He has published several books and more than 170 journal articles on the design and analysis of experiments, statistical control of stochastic processes, statistical decision theory, sequential analysis, reliability, statistical methods in logistics, and sampling from finite populations. A Fellow of the American Statistical Association, Institute of Mathematical Sciences, and American Association for the Advancement of Sciences, Dr. Zacks is the author of Stage-Wise Adaptive Designs, also published by Wiley.

Preface xv

List of Random Variables xvii

List of Abbreviations xix

1 Basic Probability Theory 1

PART I: THEORY, 1

1.1 Operations on Sets, 1

1.2 Algebra and σ-Fields, 2

1.3 Probability Spaces, 4

1.4 Conditional Probabilities and Independence, 6

1.5 Random Variables and Their Distributions, 8

1.6 The Lebesgue and Stieltjes Integrals, 12

1.7 Joint Distributions, Conditional Distributions and Independence, 21

1.8 Moments and Related Functionals, 26

1.9 Modes of Convergence, 35

1.10 Weak Convergence, 39

1.11 Laws of Large Numbers, 41

1.12 Central Limit Theorem, 44

1.13 Miscellaneous Results, 47

PART II: EXAMPLES, 56

PART III: PROBLEMS, 73

PART IV: SOLUTIONS TO SELECTED PROBLEMS, 93

2 Statistical Distributions 106

PART I: THEORY, 106

2.1 Introductory Remarks, 106

2.2 Families of Discrete Distributions, 106

2.3 Some Families of Continuous Distributions, 109

2.4 Transformations, 118

2.5 Variances and Covariances of Sample Moments, 120

2.6 Discrete Multivariate Distributions, 122

2.7 Multinormal Distributions, 125

2.8 Distributions of Symmetric Quadratic Forms of Normal Variables, 130

2.9 Independence of Linear and Quadratic Forms of Normal Variables, 132

2.10 The Order Statistics, 133

2.11 t-Distributions, 135

2.12 F-Distributions, 138

2.13 The Distribution of the Sample Correlation, 142

2.14 Exponential Type Families, 144

2.15 Approximating the Distribution of the Sample Mean: Edgeworth and Saddlepoint Approximations, 146

PART II: EXAMPLES, 150

PART III: PROBLEMS, 167

PART IV: SOLUTIONS TO SELECTED PROBLEMS, 181

3 Sufficient Statistics and the Information in Samples 191

PART I: THEORY, 191

3.1 Introduction, 191

3.2 Definition and Characterization of Sufficient Statistics, 192

3.3 Likelihood Functions and Minimal Sufficient Statistics, 200

3.4 Sufficient Statistics and Exponential Type Families, 202

3.5 Sufficiency and Completeness, 203

3.6 Sufficiency and Ancillarity, 205

3.7 Information Functions and Sufficiency, 206

3.8 The Fisher Information Matrix, 212

3.9 Sensitivity to Changes in Parameters, 214

PART II: EXAMPLES, 216

PART III: PROBLEMS, 230

PART IV: SOLUTIONS TO SELECTED PROBLEMS, 236

4 Testing Statistical Hypotheses 246

PART I: THEORY, 246

4.1 The General Framework, 246

4.2 The Neyman–Pearson Fundamental Lemma, 248

4.3 Testing One-Sided Composite Hypotheses in MLR Models, 251

4.4 Testing Two-Sided Hypotheses in One-Parameter Exponential Families, 254

4.5 Testing Composite Hypotheses with Nuisance Parameters—Unbiased Tests, 256

4.6 Likelihood Ratio Tests, 260

4.7 The Analysis of Contingency Tables, 271

4.8 Sequential Testing of Hypotheses, 275

PART II: EXAMPLES, 283

PART III: PROBLEMS, 298

PART IV: SOLUTIONS TO SELECTED PROBLEMS, 307

5 Statistical Estimation 321

PART I: THEORY, 321

5.1 General Discussion, 321

5.2 Unbiased Estimators, 322

5.3 The Efficiency of Unbiased Estimators in Regular Cases, 328

5.4 Best Linear Unbiased and Least-Squares Estimators, 331

5.5 Stabilizing the LSE: Ridge Regressions, 335

5.6 Maximum Likelihood Estimators, 337

5.7 Equivariant Estimators, 341

5.8 Estimating Equations, 346

5.9 Pretest Estimators, 349

5.10 Robust Estimation of the Location and Scale Parameters of Symmetric Distributions, 349

PART II: EXAMPLES, 353

PART III: PROBLEMS, 381

PART IV: SOLUTIONS OF SELECTED PROBLEMS, 393

6 Confidence and Tolerance Intervals 406

PART I: THEORY, 406

6.1 General Introduction, 406

6.2 The Construction of Confidence Intervals, 407

6.3 Optimal Confidence Intervals, 408

6.4 Tolerance Intervals, 410

6.5 Distribution Free Confidence and Tolerance Intervals, 412

6.6 Simultaneous Confidence Intervals, 414

6.7 Two-Stage and Sequential Sampling for Fixed Width Confidence Intervals, 417

PART II: EXAMPLES, 421

PART III: PROBLEMS, 429

PART IV: SOLUTION TO SELECTED PROBLEMS, 433

7 Large Sample Theory for Estimation and Testing 439

PART I: THEORY, 439

7.1 Consistency of Estimators and Tests, 439

7.2 Consistency of the MLE, 440

7.3 Asymptotic Normality and Efficiency of Consistent Estimators, 442

7.4 Second-Order Efficiency of BAN Estimators, 444

7.5 Large Sample Confidence Intervals, 445

7.6 Edgeworth and Saddlepoint Approximations to the Distribution of the MLE: One-Parameter Canonical Exponential Families, 446

7.7 Large Sample Tests, 448

7.8 Pitman’s Asymptotic Efficiency of Tests, 449

7.9 Asymptotic Properties of Sample Quantiles, 451

PART II: EXAMPLES, 454

PART III: PROBLEMS, 475

PART IV: SOLUTION OF SELECTED PROBLEMS, 479

8 Bayesian Analysis in Testing and Estimation 485

PART I: THEORY, 485

8.1 The Bayesian Framework, 486

8.2 Bayesian Testing of Hypothesis, 491

8.3 Bayesian Credibility and Prediction Intervals, 501

8.4 Bayesian Estimation, 502

8.5 Approximation Methods, 506

8.6 Empirical Bayes Estimators, 513

PART II: EXAMPLES, 514

PART III: PROBLEMS, 549

PART IV: SOLUTIONS OF SELECTED PROBLEMS, 557

9 Advanced Topics in Estimation Theory 563

PART I: THEORY, 563

9.1 Minimax Estimators, 563

9.2 Minimum Risk Equivariant, Bayes Equivariant, and Structural Estimators, 565

9.3 The Admissibility of Estimators, 570

PART II: EXAMPLES, 585

PART III: PROBLEMS, 592

PART IV: SOLUTIONS OF SELECTED PROBLEMS, 596

References 601

Author Index 613

Subject Index 617