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. No need to go with the expensive latest edition.

-Rachel Traylor, Ph.D.

### Review

## Prerequisites

### Differential and Integral calculus

## 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

### Attributes

Difficulty | 3 |

Good for teaching? | 5 |

Proof Quality | 4 |

Quality of Exercises | 5 |

Self-Study | 4 |

Readability | 4 |