Book:George C. Casella/Statistical Inference/Second Edition
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George C. Casella and Roger L. Berger: Statistical Inference (2nd Edition)
Published $\text {2002}$
Subject Matter
Contents
- 1 Probability Theory
- 1.1 Set Theory
- 1.2 Basics of 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 Square 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
- 10.1 Point Estimation
- 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 $x = x_0$
- 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
- Appendix: Computer Algebra
- Table of Common Distributions
- References
- Author Index
- Subject Index