Statistical Inference

This book builds theoretical statistics from the first principles of probability theory. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and are natural extensions and consequences of previous concepts. Intended for first-year graduate students, this book can be used for students majoring in statistics who have a solid mathematics background. It can also be used in a way that stresses the more practical uses of statistical theory, being more concerned with understanding basic statistical concepts and deriving reasonable statistical procedures for a variety of situations, and less concerned with formal optimality investigations.

1 Probability Theory 1 1.1 Set Theory 1 1.2 Basics of Probability Theory 5 1.2.1 Axiomatic Foundations 5 1.2.2 The Calculus of Probabilities 9 1.2.3 Counting 13 1.2.4 Enumerating Outcomes 16 1.3 Conditional Probability and Independence 20 1.4 Random Variables 27 1.5 Distribution Functions 29 1.6 Density and Mass Functions 34 1.7 Exercises 37 1.8 Miscellanea 44 2 Transformations and Expectations 47 2.1 Distributions of Functions of a Random Variable 47 2.2 Expected Values 55 2.3 Moments and Moment Generating Functions 59 2.4 Differentiating Under an Integral Sign 68 2.5 Exercises 76 2.6 Miscellanea 82 3 Common Families of Distributions 85 3.1 Introduction 85 3.2 Discrete Distributions 85 3.3 Continuous Distributions 98 3.4 Exponential Families 111 3.5 Location and Scale Families 116 3.6 Inequalities and Identities 121 3.6.1 Proability Inequalities 122 3.6.2 Identities 123 3.7 Exercises 127 3.8 Miscellanea 135 4 Multiple Random Variables 139 4.1 Joint and Marginal Distributions 139 4.2 Conditional Distributions and Independence 147 4.3 Bivariate Transformations 156 4.4 Hierarchical Models and Mixture Distributions 162 4.5 Covariance and Correlation 169 4.6 Multivariate Distributions 177 4.7 Inequalities 186 4.7.1 Numerical Inequalities 186 4.7.2 Functional Inequalities 189 4.8 Exercises 192 4.9 Miscellanea 203 5 Properties of a Random Sample 207 5.1 Basic Concepts of Random Samples 207 5.2 Sums of Random Variables from a Random Sample 211 5.3 Sampling from the Normal Distribution 218 5.3.1 Properties of the Sample Mean and Variance 218 5.3.2 The Derived Distributions: Student's t and Snedecor's F 222 5.4 Order Statistics 226 5.5 Convergence Concepts 232 5.5.1 Convergence in Probability 232 5.5.2 Almost Sure Convergence 234 5.5.3 Convergence in Distribution 235 5.5.4 The Delta Method 240 5.6 Generating a Random Sample 245 5.6.1 Direct Methods 247 5.6.2 Indirect Methods 251 5.6.3 The Accept/Reject Algorithm 253 5.7 Exercises 255 5.8 Miscellanea 267 6 Principles of Data Reduction 271 6.1 Introduction 271 6.2 The Sufficiency Principle 272 6.2.1 Sufficient Statistics 272 6.2.2 Minimal Sufficient Statistics 279 6.2.3 Ancillary Statistics 282 6.2.4 Sufficient, Ancillary, and Complete Statistics 284 6.3 The Likelihood Principle 290 6.3.1 The Likelihood Function 290 6.3.2 The Formal Likelihood Principle 292 6.4 The Equivariance Principle 296 6.5 Exercises 300 6.6 Miscellanea 307 7 Point Estimation 311 7.1 Introduction 311 7.2 Methods of Finding Estimators 312 7.2.1 Method of Moments 312 7.2.2M aximum Likelihood Estimators 315 7.2.3 Bayes Estimators 324 7.2.4 The EM Algorithm 326 7.3 Methods of Evaluating Estimators 330 7.3.2 Best Unbiased Estimators 334 7.3.3 Sufficiency and Unbiasedness 342 7.3.4 Loss Function Optimality 348 7.4 Exercises 355 7.5 Miscellanea 367 8 Hypothesis Testing 373 8.1 Introduction 373 8.2 Methods of Finding Tests 374 8.2.1 Likelihood Ratio Tests 374 8.2.2 Bayesian Tests 379 8.2.3 Union-Intersection and Intersection-Union Tests 380 8.3 Methods of Evaluating Tests 382 8.3.1 Error Probabilities and the Power Function 382 8.3.2 Most Powerful Tests 387 8.3.3 Sizes of Union-Intersection and Intersection-Union Tests 394 8.3.4 p-Values 397 8.3.5 Loss Function Optimality 400 8.4 Exercises 402 8.5 Miscellanea 413 9 Interval Estimation 417 9.1 Introduction 417 9.2 Methods of Finding Interval Estimators 420 9.2.1 Inverting a Test Statistic 420 9.2.2 Pivotal Quantities 427 9.2.3 Pivoting the CDF 430 9.2.4 Bayesian Intervals 435 9.3 Methods of Evaluating Interval Estimators 440 9.3.1 Size and Coverage Probability 440 9.3.2 Test-Related Optimality 444 9.3.3 Bayesian Optimality 447 9.3.4 Loss Function Optimality 449 9.4 Exercises 451 9.5 Miscellanea 463 10 Asymptotic Evaluations 461 10.1 Point Estimation 467 10.1.1 Consistency 467 10.1.2 Efficiency 470 10.1.3 Calculations and Comparisons 473 10.1.4 Bootstrap Standard Errors 478 10.2 Robustness 481 10.2.1 The Mean and the Median 482 10.2.2 M-Estimators 484 10.3 Hypothesis Testing 488 10.3.1 Asymptotic Distribution of LRTs 488 10.3.2 Other Large-Sample Tests 492 10.4 Interval Estimation 496 10.4.1 Approximate Maximum Likelihood Intervals 496 10.4.2 Other Large-Sample Intervals 499 10.5 Exercises 504 10.6 Miscellanea 515 11 Analysis of Variance and Regression 521 11.1 Introduction 521 11.2 Oneway Analysis of Variance 522 11.2.1 Model and Distribution Assumptions 524 11.2.2 The Classic ANOVA Hypothesis 525 11.2.3 Inferences Regarding Linear Combinations of Means 527 11.2.4 The ANOVA F Test 530 11.2.5 Simultaneous Estimation of Contrasts 534 11.2.6 Partitioning Sums of Squares 536 11.3 Simple Linear Regression 539 11.3.1 Least Squares: A Mathematical Solution 542 11.3.2 Best Linear Unbiased Estimators: A Statistical Solution 544 11.3.3 Models and Distribution Assumptions 548 11.3.4 Estimation and Testing with Normal Errors 550 11.3.5 Estimation and Prediction at a Specified x = xo 557 11.3.6 Simultaneous Estimation and Confidence Bands 559 11.4 Exercises 563 11.5 Miscellanea 572 12 Regression Models 577 12.1 Introduction 577 12.2 Regression with Errors in Variables 577 12.2.1 Functional and Structural Relationships 579 12.2.2 A Least Squares Solution 581 12.2.3 Maximum Likelihood Estimation 583 12.2.4 Confidence Sets 588 12.3 Logistic Regression 591 12.3.1 The Model 591 12.3.2 Estimation 593 12.4 Robust Regression 597 12.5 Exercises 602 12.B Miscellanea 608 Appendix: Computer Algebra 613 Table of Common Distributions 621 References 629 Author Index 645 Subject Index 649

Casella 在福特汉姆大学完成了他的本科教育，在普渡大学完成了研究生教育。他曾在罗格斯大学、康奈尔大学和佛罗里达大学任教。他的贡献集中在统计学领域，包括蒙特卡洛方法、模型选择和基因组分析。他在贝叶斯和经验贝叶斯方法方面特别活跃，其作品与斯坦因现象相联系，涉及评估和加速马尔科夫链蒙特卡洛方法的收敛性，如他的Rao-Blackwellisation技术，以及将拉索重塑为具有独立拉普拉斯先验的贝叶斯后验模式估计。 Casella于1988年被任命为美国统计学会和数学统计研究所的研究员，并于1989年成为国际统计学会的当选研究员。2009年，他被任命为西班牙皇家科学院外籍院士。 Roger Lee Berger 是美国统计学家和教授，与合作者 George Casella 于1990年首次出版《Statistical Inference》一书。

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