Review: Introduction to Mathematical Statistics (Hogg et al)

The Math Citadel

ISBN

0-13-008507-3

Classification: Statistics and Data Science

Hathlor code:M-STS.10011110

Prerequisites

Differential and Integral calculus

Ranking

All ranking is done on a scale of 1-5, with 5 as the best.

Attribute	Rank
Difficulty	3
Proof Quality	4
Readability	4
Self-Study	4
Good for Teaching?	5
Quality of Exercises	5

Topics Covered

Basic probability
- Conditional probability, independence
- discrete and continuous random variables, various distributions
- expectation of random variables
- important inequalities (e.g. Chebyshev's inequality)
Multivarite Distributions
- Bivariate distributions, marginal distributions, conditional distributions
- correlation coefficient
Special Distributions
- Binomial, Multinomial
- Poisson
- $\Gamma, \chi^{2}, \beta$ distributions
- Normal and multivariate normal
- t and F distributions
Consistency, Bias, and Limiting Distributions
- Convergence in probability and distribution
- central limit theorem
Elementary Statistical Inference
- Order statistics
- confidence intervals
- hypothesis testing
- Chi-Square tests
- Monte-Carlo methods
- Bootstrapping
Maximum Likelihood Methods
Sufficient Statistics
Optimal Hypothesis Testing
Inference on Normal Models
- One-way ANOVA
- noncentral $\chi^{2}$ and F distributions
- Multiple comparisons
Nonparametric Statistics
- Sign test
- Signed-Rank Wilcoxon test
- Mann-Whitney-Wilcoxon procedure
- general rank tests
- nonparametric measures of association
Bayesian Statistics
Linear Models
- Robustness

Review

"For a good introduction to the mathematics of statistics, I can think of few books I would recommend as highly as this book by Hogg, McKean, and Craig. The authors do not assume any familiarity with statistics or probability, yet treat the student to a full development of the subject. Proofs are included and well written and informative. The exercises build a good foundation for the reader; some require proofs to develop further understanding, some are derivations, and some are specific examples.

The topics covered are broad, and each topic is accompanied by numerous illuminating examples to give a reader an almost tactile look at a topic as they learn to treat it abstractly. For a simple illustration, they take the time to give several examples to illustrate what the intersection of sets look like, both in discrete and continuous space. They also take the time to supplement with figures to aid in visualization, which isn't common in mathematical statistics books.

Most of this book focuses on basic probability, statistical inference (mostly parametric, but some nonparametric as well), and has a little bit of introductory Bayesian statistics thrown in.

The book would be great for self-study, provided the reader has had calculus and is comfortable with it. The calculus is fairly simple, but a solid grasp of it will make this text much easier to study. I would recommend this book for teaching a two semester undergraduate or a one-semester graduate course in introductory mathematical statistics.

Note: The edition I am reviewing is the 6th edition. Later editions will cost a great deal more money, and this edition is perfectly sufficient. There is no need to go with the expensive latest edition.