Introduction to Mathematical Statistics, Sixth Edition

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Introduction to Mathematical Statistics, Sixth Edition

ISBN-13: 978-0-13-008507-8 See more

Author(s): Robert V. Hogg; Joseph W. McKean; Allen T. Craig

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Introduction to Mathematical Statistics, Seventh Edition

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ISBN: 9780321795441

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eText:

ISBN-10 0-321-72489-5, ISBN-13 978-0-321-72489-2

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ISBN-10 0-13-008507-3, ISBN-13 978-0-13-008507-8

Author(s): Robert V. Hogg; Joseph W. McKean; Allen T. Craig

Publisher: Pearson

Description

For one or two-semester, undergraduate mathematical statistics course, or for beginning graduate courses in mathematical statistics.

This classic text retains its outstanding features and continues to provide students with excellent background in the mathematics of statistics. Extensively revised with three new chapters.

Preface, xi

1. Probability and Distributions, 1

1.1. Introduction, 1

1.2. Set Theory, 3

1.3. The Probability Set Function, 11

1.4. Conditional Probability and Independence, 22

1.5. Random Variables, 33

1.6. Discrete Random Variables, 41

1.6.1. Transformations, 43

1.7. Continuous Random Variables, 45

1.7.1. Transformations, 47

1.8. Expectation of a Random Variable, 53

1.9. Some Special Expectations, 59

1.10. Important Inequalities, 68

2. Multivariate Distributions, 73

2.1. Distributions of Two Random Variables, 73

2.1.1. Expectation, 79

2.2. Transformations: Bivariate Random Variables, 84

2.3. Conditional Distributions and Expectations, 93

2.4. The Correlation Coefficient, 101

2.5. Independent Random Variables, 108

2.6. Extension to Several Random Variables, 115

2.6.1. *Variance-Covariance, 121

2.7. Transformations: Random Vectors, 124

3. Some Special Distributions, 133

3.1. The Binomial and Related Distributions, 133

3.2. The Poisson Distribution, 143

3.3. The Γ, χ², and β Distributions, 149

3.4. The Normal Distribution, 160

3.4.1. Contaminated Normals, 167

3.5. The Multivariate Normal Distribution, 171

3.5.1. *Applications, 177

3.6. t and F-Distributions, 182

3.6.1. The t-distribution, 182

3.6.2. The F-distribution, 184

3.6.3. Student’s Theorem, 186

3.7. Mixture Distributions, 189

4. Unbiasedness, Consistency, and Limiting Distributions, 197

4.1. Expectations of Functions, 198

4.2. Convergence in Probability, 203

4.3. Convergence in Distribution, 207

4.3.1. Bounded in Probability, 213

4.3.2. Δ-Method, 214

4.3.3. Moment Generating Function Technique, 216

4.4. Central Limit Theorem, 220

4.5. *Asymptotics for Multivariate Distributions, 226

5. Some Elementary Statistical Inferences, 233

5.1. Sampling and Statistics, 233

5.2. Order Statistics, 238

5.2.1. Quantiles, 242

5.2.2. Confidence Intervals of Quantiles, 245

5.3. *Tolerance Limits for Distributions, 250

5.4. More on Confidence Intervals, 254

5.4.1. Confidence Intervals for Differences in Means, 257

5.4.2. Confidence Interval for Difference in Proportions, 260

5.5. Introduction to Hypothesis Testing, 263

5.6. Additional Comments About Statistical Tests, 272

5.7. Chi-Square Tests, 278

5.8. The Method of Monte Carlo, 286

5.8.1. Accept-Reject Generation Algorithm, 292

5.9. Bootstrap Procedures, 297

5.9.1. Percentile Bootstrap Confidence Intervals, 297

5.9.2. Bootstrap Testing Procedures, 301

6. Maximum Likelihood Methods, 311

6.1. Maximum Likelihood Estimation, 311

6.2. Rao-Cramér Lower Bound and Efficiency, 319

6.3. Maximum Likelihood Tests, 333

6.4. Multiparameter Case: Estimation, 342

6.5. Multiparameter Case: Testing, 351

6.6. The EM Algorithm, 359

7. Sufficiency, 367

7.1. Measures of Quality of Estimators, 367

7.2. A Sufficient Statistic for a Parameter, 373

7.3. Properties of a Sufficient Statistic, 380

7.4. Completeness and Uniqueness, 385

7.5. The Exponential Class of Distributions, 389

7.6. Functions of a Parameter, 394

7.7. The Case of Several Parameters, 398

7.8. Minimal Sufficiency and Ancillary Statistics, 406

7.9. Sufficiency, Completeness and Independence, 411

8. Optimal Tests of Hypotheses, 419

8.1. Most Powerful Tests, 419

8.2. Uniformly Most Powerful Tests, 429

8.3. Likelihood Ratio Tests, 437

8.4. The Sequential Probability Ratio Test, 448

8.5. Minimax and Classification Procedures, 455

8.5.1. Minimax Procedures, 456

8.5.2. Classification, 458

9. Inferences about Normal Models, 463

9.1. Quadratic Forms, 463

9.2. One-way ANOVA, 468

9.3. Noncentral χ² and F Distributions, 475

9.4. Multiple Comparisons, 477

9.5. The Analysis of Variance, 482

9.6. A Regression Problem, 488

9.7. A Test of Independence, 498

9.8. The Distributions of Certain Quadratic Forms, 501

9.9. The Independence of Certain Quadratic Forms, 508

10. Nonparametric Statistics, 515

10.1. Location Models, 515

10.2. Sample Median and Sign Test, 518

10.2.1. Asymptotic Relative Efficiency, 523

10.2.2. Estimating Equations Based on Sign Test, 528

10.2.3. Confidence Interval for the Median, 529

10.3. Signed-Rank Wilcoxon, 531

10.3.1. Asymptotic Relative Efficiency, 536

10.3.2. Estimating Equations Based on Signed-rank Wilcoxon, 539

10.3.3. Confidence Interval for the Median, 539

10.4. Mann-Whitney-Wilcoxon Procedure, 541

10.4.1. Asymptotic Relative Efficiency, 545

10.4.2. Estimating Equations Based on the Mann-Whitney-Wilcoxon, 547

10.4.3. Confidence Interval for the Shift Parameter Δ, 547

10.5. General Rank Scores, 549

10.5.1. Efficacy, 552

10.5.2. Estimating Equations Based on General Scores, 553

10.5.3. Optimization: Best Estimates, 554

10.6. Adaptive Procedures, 561

10.7. Simple Linear Model, 565

10.8. Measures of Association, 570

10.8.1. Kendall’s τ, 571

10.8.2. Spearman’s Rho, 574

11. Bayesian Statistics, 579

11.1. Subjective Probability, 579

11.2. Bayesian Procedures, 582

11.2.1. Prior and Posterior Distributions, 583

11.2.2. Bayesian Point Estimation, 586

11.2.3. Bayesian Interval Estimation, 589

11.2.4. Bayesian Testing Procedures, 590

11.2.5. Bayesian Sequential Procedures, 592

11.3. More Bayesian Terminology and Ideas, 593

11.4. Gibbs Sampler, 600

11.5. Modern Bayesian Methods, 606

11.5.1. Empirical Bayes, 610

12. Linear Models, 615

12.1. Robust Concepts, 615

12.1.1. Norms and Estimating Equations, 616

12.1.2. Influence Functions, 617

12.1.3. Breakdown Point of an Estimator, 621

12.2. LS and Wilcoxon Estimators of Slope, 624

12.2.1. Norms and Estimating Equations, 625

12.2.2. Influence Functions, 626

12.2.3. Intercept, 629

12.3. LS Estimation for Linear Models, 631

12.3.1. Least Squares, 633

12.3.2. Basics of LS Inference under Normal Errors, 635

12.4. Wilcoxon Estimation for Linear Models, 640

12.4.1. Norms and Estimating Equations, 641

12.4.2. Influence Functions, 641

12.4.3. Asymptotic Distribution Theory, 643

12.4.4. Estimates of the Intercept Parameter, 645

12.5. Tests of General Linear Hypotheses, 647

12.5.1. Distribution Theory for the LS Test for Normal Errors, 650

12.5.2. Asymptotic Results, 651

12.5.3. Examples, 652

Appendix A. Mathematics, 661

A.1. Regularity Conditions, 661

A.2. Sequences, 662

Appendix B. R and S-PLUS Functions, 665

Appendix C. Tables of Distributions, 671

Appendix D. References, 679

Appendix E. Answers to Selected Exercises, 683

Index, 693

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Introduction to Mathematical Statistics, Sixth Edition

ISBN-13: 978-0-13-008507-8 See more

Author(s): Robert V. Hogg; Joseph W. McKean; Allen T. Craig

Price Information

Additional product details

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