Statistical Inference

Citation

Casella, G., & Berger, R. L. (2024). Statistical Inference (2nd ed.). CRC Press. (Reprint of the second edition first published by Cengage Learning in 2001.)

Chapter Summary

The book builds theoretical statistics from the first principles of probability theory. The authors develop the theory of inference through techniques, definitions and concepts that are natural extensions of those that precede them. The first five chapters are probability theory; the remaining seven are inference proper. The edition described here has twelve chapters, an appendix on computer algebra, a table of common distributions and references.

Chapter 1: Probability Theory

The starting point of the book, which develops probability from first principles. The first five chapters together cover probability theory, including multivariate distributions, limit theorems and random samples.

Chapter 2: Transformations and Expectations

How the distribution and expected values of a random variable change when it is transformed.

Chapter 3: Common Families of Distributions

The families of distributions that serve as models throughout the inference chapters. The book closes with a table of common distributions for reference.

Chapter 4: Multiple Random Variables

The theory of several random variables considered jointly, which statistics needs because data are samples of many observations.

Chapter 5: Properties of a Random Sample

What can be said about samples and the statistics computed from them before any inference is attempted, the bridge from probability to statistics.

Chapter 6: Principles of Data Reduction

The ways in which data can be summarised without losing the information relevant to a parameter. A review of the first edition names sufficiency, likelihood and invariance among them.

Chapter 7: Point Estimation

Methods for constructing estimators of a parameter and criteria for judging them.

Chapter 8: Hypothesis Testing

The formulation and evaluation of tests of hypotheses about parameters.

Chapter 9: Interval Estimation

Procedures that produce an interval of plausible parameter values in place of a single number.

Chapter 10: Asymptotic Evaluations

Properties of estimators and procedures when the sample size grows large.

Chapters 11–12: Analysis of Variance and Regression; Regression Models

Chapter 11 applies the machinery of inference to analysis of variance and regression, and Chapter 12 to regression models.

Key Concepts

Chapters 1–5: Probability Foundations

• Transformations of random variables: finding the distribution of a function of a random variable.
• Joint distributions: describing several random variables together.
• Random sample: independent observations from a common distribution and the statistics computed from them.

Chapters 6–10: Theory of Inference

• Data reduction: summarising a sample without losing information about the parameter (sufficiency).
• Point and interval estimation: constructing and evaluating estimators and interval procedures.
• Hypothesis testing: procedures for deciding between competing statements about a parameter.
• Asymptotic evaluation: studying estimators as the sample size grows.

Chapters 11–12: Linear Models

• Analysis of variance and regression: inference for comparisons among groups and for relationships between variables.

Critical Analysis

Strengths

• A theory built from probability up. The publisher describes the book as developing the theory of inference through techniques and concepts that follow from previous ones, covering all the topics of a standard inference course: distributions, random variables, data reduction, point estimation, hypothesis testing and interval estimation.
• Accessible to anyone with some calculus. The publisher says the style is lucid and requires only some background in calculus, with hundreds of examples and extensive sets of graduated exercises in every chapter.
• An emphasis on understanding. The publisher says the book is concerned with understanding basic statistical concepts and deriving reasonable procedures, and is less focused on formal optimality considerations.
• Reviewed as clearly written. Lionel Weiss, reviewing the first edition for zbMATH, called the book very clearly written, with many exercises and material not usually found in an introductory textbook, such as Stein's phenomenon.

Limitations

• Graduate-level demand. The publisher says the book is primarily aimed at graduate students of statistics and can be used by advanced undergraduates only with a solid mathematics background.
• Optimality theory is not the focus. Readers who want a deeper treatment of optimal estimation and testing will need a further text.
• A reprint of a 2001 edition. The 2024 CRC volume reproduces the second edition from 2001, so it does not include later developments in the field.
• Changed structure. The first edition had a decision theory chapter in its list of chapters; the twelve chapters of the second edition do not. Readers moving between editions should compare the contents.

Where It Fits in the Course

This is the standard-theory book of the course: calculus-based, built from probability, covering estimation, testing and linear models. Joseph Blitzstein and Jessica Hwang's Introduction to Probability is a good preparation for the first five chapters. An Introduction to Mathematical Statistics and Its Applications covers probability, estimation, testing, regression and analysis of variance, and so overlaps much of the same ground. David Spiegelhalter's The Art of Statistics and the free Introductory Statistics 2e are non-technical or applied and complement the theory rather than compete with it.

David Williams's Probability with Martingales supplies the measure-theoretic footing that Casella and Berger do without. Probability and Statistical Inference by Mavrakakis and Penzer is a graduate-level book that also includes Bayesian methods and simulation, which are not named in the chapter titles here. Mitzenmacher and Upfal's Probability and Computing addresses computer science rather than statistics.

How to Use This Book

The book is for students of statistics beginning graduate study and for advanced undergraduates with a solid mathematics background, including a year of calculus.

If probability is fresh, read Chapters 1 to 5 closely, since later chapters rely on them; if not, skim Chapters 1 to 4 and read Chapter 5 on random samples carefully. Chapters 6 to 10 are the core of inference and are best read in order: data reduction, then point estimation, hypothesis testing and interval estimation, then asymptotics. Chapters 11 and 12 can be read separately by anyone who needs analysis of variance or regression.

Work the exercises, which are graded in difficulty, and use the table of common distributions as a reference. For topics outside the contents, such as Bayesian inference, take a companion text.