Statistical Inference (Chapman & Hall/CRC Texts in Statistical Science)

✍ Scribed by George Casella, Roger Berger

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 566
• Edition: 2
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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