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This classic textbook 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 natural extensions, and consequences, of previous concepts. It covers all topics from a standard inference course including: distributions, random variables, data reduction, point estimation, hypothesis testing, and interval estimation. Features The classic graduate-level textbook on statistical inference Develops elements of statistical theory from first principles of probability Written in a lucid style accessible to anyone with some background in calculus Covers all key topics of a standard course in inference Hundreds of examples throughout to aid understanding Each chapter includes an extensive set of graduated exercises Statistical Inference, Second Edition is primarily aimed at graduate students of statistics, but can be used by advanced undergraduate students majoring in statistics who have a solid mathematics background. It also stresses the more practical uses of statistical theory, being more concerned with understanding basic statistical concepts and deriving reasonable statistical procedures, while less focused on formal optimality considerations. This is a reprint of the second edition originally published by Cengage Learning, Inc. in 2001. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface to the Second Edition Preface to the First Edition Authors List of Tables List of Figures List of Examples 1. Probability Theory 1.1. Set Theory 1.2. Probability Theory 1.2.1. Axiomatic Foundations 1.2.2. The Calculus of Probabilities 1.2.3. Counting 1.2.4. Enumerating Outcomes 1.3. Conditional Probability and Independence 1.4. Random Variables 1.5. Distribution Functions 1.6. Density and Mass Functions 1.7. Exercises 1.8. Miscellanea 2. Transformations and Expectations 2.1. Distributions of Functions of a Random Variable 2.2. Expected Values 2.3. Moments and Moment Generating Functions 2.4. Differentiating under an Integral Sign 2.5. Exercises 2.6. Miscellanea 3. Common Families of Distributions 3.1. Introduction 3.2. Discrete Distributions 3.3. Continuous Distributions 3.4. Exponential Families 3.5. Location and Scale Families 3.6. Inequalities and Identities 3.6.1. Probability Inequalities 3.6.2. Identities 3.7. Exercises 3.8. Miscellanea 4. Multiple Random Variables 4.1. Joint and Marginal Distributions 4.2. Conditional Distributions and Independence 4.3. Bivariate Transformations 4.4. Hierarchical Models and Mixture Distributions 4.5. Covariance and Correlation 4.6. Multivariate Distributions 4.7. Inequalities 4.7.1. Numerical Inequalities 4.7.2. Functional Inequalities 4.8. Exercises 4.9. Miscellanea 5. Properties of a Random Sample 5.1. Basic Concepts of Random Samples 5.2. Sums of Random Variables from a Random Sample 5.3. Sampling from the Normal Distribution 5.3.1. Properties of the Sample Mean and Variance 5.3.2. The Derived Distributions: Student's t and Snedecor's F 5.4. Order Statistics 5.5. Convergence Concepts 5.5.1. Convergence in Probability 5.5.2. Almost Sure Convergence 5.5.3. Convergence in Distribution 5.5.4. The Delta Method 5.6. Generating a Random Sample 5.6.1. Direct Methods 5.6.2. Indirect Methods 5.6.3. The Accept/Reject Algorithm 5.7. Exercises 5.8. Miscellanea 6. Principles of Data Reduction 6.1. Introduction 6.2. The Sufficiency Principle 6.2.1. Sufficient Statistics 6.2.2. Minimal Sufficient Statistics 6.2.3. Ancillary Statistics 6.2.4. Sufficient, Ancillary, and Complete Statistics 6.3. The Likelihood Principle 6.3.1. The Likelihood Function 6.3.2. The Formal Likelihood Principle 6.4. The Equivariance Principle 6.5. Exercises 6.6. Miscellanea 7. Point Estimation 7.1. Introduction 7.2. Methods of Finding Estimators 7.2.1. Method of Moments 7.2.2. Maximum Likelihood Estimators 7.2.3. Bayes Estimators 7.2.4. The EM Algorithm 7.3. Methods of Evaluating Estimators 7.3.1. Mean Squared Error 7.3.2. Best Unbiased Estimators 7.3.3. Sufficiency and Unbiasedness 7.3.4. Loss Function Optimality 7.4. Exercises 7.5. Miscellanea 8. Hypothesis Testing 8.1. Introduction 8.2. Methods of Finding Tests 8.2.1. Likelihood Ratio Tests 8.2.2. Bayesian Tests 8.2.3. Union–Intersection and Intersection–Union Tests 8.3. Methods of Evaluating Tests 8.3.1. Error Probabilities and the Power Function 8.3.2. Most Powerful Tests 8.3.3. Sizes of Union–Intersection and Intersection–Union Tests 8.3.4. p-Values 8.3.5. Loss Function Optimality 8.4. Exercises 8.5. Miscellanea 9. Interval Estimation 9.1. Introduction 9.2. Methods of Finding Interval Estimators 9.2.1. Inverting a Test Statistic 9.2.2. Pivotal Quantities 9.2.3. Pivoting the CDF 9.2.4. Bayesian Intervals 9.3. Methods of Evaluating Interval Estimators 9.3.1. Size and Coverage Probability 9.3.2. Test-Related Optimality 9.3.3. Bayesian Optimality 9.3.4. Loss Function Optimality 9.4. Exercises 9.5. Miscellanea 10. Asymptotic Evaluations 10.1. Point Estimation 10.1.1. Consistency 10.1.2. Efficiency 10.1.3. Calculations and Comparisons 10.1.4. Bootstrap Standard Errors 10.2. Robustness 10.2.1. The Mean and the Median 10.2.2. M-Estimators 10.3. Hypothesis Testing 10.3.1. Asymptotic Distribution of LRTs 10.3.2. Other Large-Sample Tests 10.4. Interval Estimation 10.4.1. Approximate Maximum Likelihood Intervals 10.4.2. Other Large-Sample Intervals 10.5. Exercises 10.6. Miscellanea 11. Analysis of Variance and Regression 11.1. Introduction 11.2. Oneway Analysis of Variance 11.2.1. Model and Distribution Assumptions 11.2.2. The Classic ANOVA Hypothesis 11.2.3. Inferences Regarding Linear Combinations of Means 11.2.4. The ANOVA F Test 11.2.5. Simultaneous Estimation of Contrasts 11.2.6. Partitioning Sums of Squares 11.3. Simple Linear Regression 11.3.1. Least Squares: A Mathematical Solution 11.3.2. Best Linear Unbiased Estimators: A Statistical Solution 11.3.3. Models and Distribution Assumptions 11.3.4. Estimation and Testing with Normal Errors 11.3.5. Estimation and Prediction at a Specified x = x0 11.3.6. Simultaneous Estimation and Confidence Bands 11.4. Exercises 11.5. Miscellanea 12. Regression Models 12.1. Introduction 12.2. Regression with Errors in Variables 12.2.1. Functional and Structural Relationships 12.2.2. A Least Squares Solution 12.2.3. Maximum Likelihood Estimation 12.2.4. Confidence Sets 12.3. Logistic Regression 12.3.1. The Model 12.3.2. Estimation 12.4. Robust Regression 12.5. Exercises 12.6. Miscellanea Computer Algebra Table of Common Distributions References Index