Introduction to Mathematical Statistics

(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 ?, ?2, and ? 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-Cramer 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 ?2 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:




A. Mathematical Comments

A.1 Regularity Conditions

A.2 Sequences




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




C. Lists of Common Distributions




D. Table of Distributions




E. References




F. Answers to Selected Exercises




Index