Introduction to Mathematical Statistics

Chapter 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

Chapter 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

Chapter 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*

Chapter 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*

Chapter 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*

Chapter 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

Chapter 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

Chapter 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*

Chapter 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

Chapter 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

Chapter 11: Bayesian Statistics

• • 11.1 Bayesian Procedures
• • 11.2 More Bayesian Terminology and Ideas
• • 11.3 Gibbs Sampler
• • 11.4 Modern Bayesian Methods

Chapter A: Mathematical Comments

• • A.1 Regularity Conditions
• • A.2 Sequences

Chapter B: R Primer

• • B.1 Basics
• • B.2 Probability Distributions
• • B.3 R Functions
• • B.4 Loops
• • B.5 Input and Output
• • B.6 Packages

Chapter C: Lists of Common Distributions

Chapter D: Table of Distributions

Chapter E: References

Chapter F: Answers to Selected Exercises