Introduction to Mathematical Statistics

By Robert V. Hogg, Joseph W. McKean and Allen T. Craig · Pearson · 2018

A comprehensive textbook of mathematical statistics that treats classical inference procedures in estimation and testing extensively. In the publisher's description it promotes understanding through numerous illustrative examples and exercises, and its flexible organization suits a range of mathematical statistics courses. The 8th edition's revisions connect statistical theory with statistical practice: use of the statistical software R is expanded throughout, more real data sets (in the free R package hmcpkg) illustrate or compare methods, and new topics include Tukey's multiple comparison procedure in Chapter 9 and confidence intervals for correlation coefficients in Chapters 9 and 10.

On the syllabus

Exam MAS-I — objectives B1–B8 and C1–C9; for Domain B, Sections 1.7, 2.7, 2.8, 3.7, 4.1, 4.4, 4.5, 4.6, 4.7, 5.1, 6.1, 6.2, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2 and 8.3; for Domain C, Sections 4.4, 6.3, 6.5, 7.5, 9.1, 9.2, 9.3, 9.4, 9.5 and 9.6.

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 $\Gamma$ , $\chi^2$ , and $\beta$ 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
8.4 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 $\chi^2$ 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

Appendix A Mathematical Comments

A.1 Regularity Conditions
A.2 Sequences

Appendix 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

Appendix C Lists of Common Distributions

Appendix D Tables of Distributions

Appendix E References

Appendix F Answers to Selected Exercises

Index

Sources

Introduction to Mathematical Statistics, 8th edition (Pearson, 2018) — the publisher's description, the table of contents (chapters, sections and appendices) and the hardcover ISBN 978-0-13-468699-8, published 2018
Introduction to Mathematical Statistics, 8th edition, Global Edition — contents pages (Pearson, 2020) — a library scan of the printed contents pages, which agree section for section with the publisher's list; the appendix letters A–F and the titles of Appendices C–F are taken from it
CAS Exam MAS-I Content Outline (2026) — the citation (8th edition, Prentice Hall, 2018) and the assigned sections by domain