内容

目次

Preface to the Second Edition xi Preface to the First Edition xiii 1. Probability 1 1.1 Introduction 1 1.2 Sample Space 2 1.3 Probability Axioms 7 1.4 Combinatorics: Probability on Finite Sample Spaces 21 1.5 Conditional Probability and Bayes Theorem 28 1.6 Independence of Events 33 2. Random Variables and Their Probability Distributions 40 2.1 Introduction 40 2.2 Random Variables 40 2.3 Probability Distribution of a Random Variable 43 2.4 Discrete and Continuous Random Variables 48 2.5 Functions of a Random Variable 57 3. Moments and Generating Functions 69 3.1 Introduction 69 3.2 Moments of a Distribution Function 69 3.3 Generating Functions 85 3.4 Some Moment Inequalities 95 4. Multiple Random Variables 102 4.1 Introduction 102 4.2 Multiple Random Variables 102 4.3 Independent Random Variables 119 4.4 Functions of Several Random Variables 127 4.5 Covariance Correlation and Moments 149 4.6 Conditional Expectation 164 4.7 Order Statistics and Their Distributions 171 5. Some Special Distributions 180 5.1 Introduction 180 5.2 Some Discrete Distributions 180 5.3 Some Continuous Distributions 204 5.4 Bivariate and Multivariate Normal Distributions 238 5.5 Exponential Family of Distributions 251 6. Limit Theorems 256 6.1 Introduction 256 6.2 Modes of Convergence 256 6.3 Weak Law of Large Numbers 274 6.4 Strong Law of Large Numbers 281 6.5 Limiting Moment Generating Functions 289 6.6 Central Limit Theorem 293 7. Sample Moments and Their Distributions 306 7.1 Introduction 306 7.2 Random Sampling 307 7.3 Sample Characteristics and Their Distributions 310 7.4 Chi–Square f– and F–Distributions: Exact Sampling Distributions 324 7.5 Large–Sample Theory 334 7.6 Distribution of (X S2) in Sampling from a Normal Population 339 7.7 Sampling from a Bivariate Normal Distribution 344 8. Parametric Point Estimation 353 8.1 Introduction 353 8.2 Problem of Point Estimation 354 8.3 Sufficiency Completeness and Ancillarity 358 8.4 Unbiased Estimation 377 8.5 Unbiased Estimation (Continued): Lower Bound for the Variance of an Estimator 391 8.6 Substitution Principle (Method of Moments) 406 8.7 Maximum Likelihood Estimators 409 8.8 Bayes and Minimax Estimation 424 8.9 Principle of Equivariance 442 9. Neyman–Pearson Theory of Testing of Hypotheses 454 9.1 Introduction 454 9.2 Some Fundamental Notions of Hypotheses Testing 454 9.3 Neyman–Pearson Lemma 464 9.4 Families with Monotone Likelihood Ratio 472 9.5 Unbiased and Invariant Tests 479 9.6 Locally Most Powerful Tests 486 10. Some Further Results of Hypothesis Testing 490 10.1 Introduction 490 10.2 Generalized Likelihood Ratio Tests 490 10.3 Chi–Square Tests 500 10.4 /–Tests 512 10.5 F–Tests 518 10.6 Bayes and Minimax Procedures 520 11. Confidence Estimation 527 11.1 Introduction 527 11.2 Some Fundamental Notions of Confidence Estimation 527 11.3 Methods of Finding Confidence Intervals 532 11.4 Shortest–Length Confidence Intervals 546 11.5 Unbiased and Equivariant Confidence Intervals 553 12. General Linear Hypothesis 561 12.1 Introduction 561 12.2 General Linear Hypothesis 561 12.3 Regression Model 569 12.4 One–Way Analysis of Variance 577 12.5 Two–Way Analysis of Variance with One Observation per Cell 583 12.6 Two–Way Analysis of Variance with Interaction 590 13. Nonparametric Statistical Inference 598 13.1 Introduction 598 13.2 U–Statistics 598 13.3 Some Single–Sample Problems 608 13.4 Some Two–Sample Problems 624 13.5 Tests of Independence 633 13.6 Some Applications of Order Statistics 644 13.7 Robustness 650 References 663 Frequently Used Symbols and Abbreviations 669 Statistical Tables 673 Answers to Selected Problems 693 Author Index 705 Subject Index 707