Introduction to Mathematical Statistics, 8th edition

• (Note: Sections marked with an asterisk * are optional.)

1. Probability and Distributions
   • 1.1 Introduction
   • 1.2 Sets
   • 1.3 The Probability Set Function
   • 1.4 Conditional Probability and Independence
   • 1.5 Random Variables
   • 1.6 Discrete Random Variables
   • 1.7 Continuous Random Variables
   • 1.8 Expectation of a Random Variable
   • 1.9 Some Special Expectations
   • 1.10 Important Inequalities
2. Multivariate Distributions
   • 2.1 Distributions of Two Random Variables
   • 2.2 Transformations: Bivariate Random Variables
   • 2.3 Conditional Distributions and Expectations
   • 2.4 Independent Random Variables
   • 2.5 The Correlation Coefficient
   • 2.6 Extension to Several Random Variables
   • 2.7 Transformations for Several Random Variables
   • 2.8 Linear Combinations of Random Variables
3. Some Special Distributions
   • 3.1 The Binomial and Related Distributions
   • 3.2 The Poisson Distribution
   • 3.3 The Γ, χ2, and β Distributions
   • 3.4 The Normal Distribution
   • 3.5 The Multivariate Normal Distribution
   • 3.6 t- and F-Distributions
   • 3.7 Mixture Distributions*
4. Some Elementary Statistical Inferences
   • 4.1 Sampling and Statistics
   • 4.2 Confidence Intervals
   • 4.3 ∗Confidence Intervals for Parameters of Discrete Distributions
   • 4.4 Order Statistics
   • 4.5 Introduction to Hypothesis Testing
   • 4.6 Additional Comments About Statistical Tests
   • 4.7 Chi-Square Tests
   • 4.8 The Method of Monte Carlo
   • 4.9 Bootstrap Procedures
   • 4.10 Tolerance Limits for Distributions*
5. Consistency and Limiting Distributions
   • 5.1 Convergence in Probability
   • 5.2 Convergence in Distribution
   • 5.3 Central Limit Theorem
   • 5.4 Extensions to Multivariate Distributions*
6. Maximum Likelihood Methods
   • 6.1 Maximum Likelihood Estimation
   • 6.2 Rao—Cramér Lower Bound and Efficiency
   • 6.3 Maximum Likelihood Tests
   • 6.4 Multiparameter Case: Estimation
   • 6.5 Multiparameter Case: Testing
   • 6.6 The EM Algorithm
7. Sufficiency
   • 7.1 Measures of Quality of Estimators
   • 7.2 A Sufficient Statistic for a Parameter
   • 7.3 Properties of a Sufficient Statistic
   • 7.4 Completeness and Uniqueness
   • 7.5 The Exponential Class of Distributions
   • 7.6 Functions of a Parameter
   • 7.7 The Case of Several Parameters
   • 7.8 Minimal Sufficiency and Ancillary Statistics
   • 7.9 Sufficiency, Completeness, and Independence
8. Optimal Tests of Hypotheses
   • 8.1 Most Powerful Tests
   • 8.2 Uniformly Most Powerful Tests
   • 8.3 Likelihood Ratio Tests
   • 8.3.2 Likelihood Ratio Tests for Testing Variances of Normal Distributions
   • 8.4 The Sequential Probability Ratio Test*
   • 8.5 Minimax and Classification Procedures*
9. Inferences About Normal Linear Models
   • 9.1 Introduction
   • 9.2 One-Way ANOVA
   • 9.3 Noncentral χ2 and F-Distributions
   • 9.4 Multiple Comparisons
   • 9.5 Two-Way ANOVA
   • 9.6 A Regression Problem
   • 9.7 A Test of Independence
   • 9.8 The Distributions of Certain Quadratic Forms
   • 9.9 The Independence of Certain Quadratic Forms
10. Nonparametric and Robust Statistics
    • 10.1 Location Models
    • 10.2 Sample Median and the Sign Test
    • 10.3 Signed-Rank Wilcoxon
    • 10.4 Mann—Whitney—Wilcoxon Procedure
    • 10.5 General Rank Scores*
    • 10.6 Adaptive Procedures*
    • 10.7 Simple Linear Model
    • 10.8 Measures of Association
    • 10.9 Robust Concepts
11. Bayesian Statistics
    • 11.1 Bayesian Procedures
    • 11.2 More Bayesian Terminology and Ideas
    • 11.3 Gibbs Sampler
    • 11.4 Modern Bayesian Methods

Appendices:

1. Mathematical Comments
   • A.1 Regularity Conditions
   • A.2 Sequences
2. R Primer
   • B.1 Basics
   • B.2 Probability Distributions
   • B.3 R Functions
   • B.4 Loops
   • B.5 Input and Output
   • B.6 Packages
3. Lists of Common Distributions
4. Table of Distributions
5. References
6. Answers to Selected Exercises

Index