Book:George C. Casella/Statistical Inference/Second Edition

George C. Casella and Roger L. Berger: Statistical Inference (2nd Edition)

Published $\text {2002}$

Subject Matter

• Statistics

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

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

Further Editions

• 1990: George C. Casella and Roger L. Berger: Statistical Inference