Statistical Inference (Casella & Berger): A Chapter-by-Chapter Guide

What the book covers, how its chapters fit together, and how this four-part summary works.

Statistical Inference by George Casella and Roger L. Berger is a book I had been trying to read for a year, but there were always other priorities. When I finally had the time, I crashed against a wall of statistical theory: some problems are fun, while others are a nightmare. Don’t get me wrong, it’s a very comprehensive book about statistics in general, but sometimes it can be a little bit too heavy. Still, if you’ve ever fit a regression, read a p-value or built a confidence interval, this is the book that explains why those things work, and when they don’t.

I read it with a pen in hand and wrote a one-page summary of every chapter. This series is those notes, cleaned up and with the gaps filled in from the book. It’s a pity that I couldn’t include the mental struggle I had while writing those summaries.

Where I am after this book: still in the middle of the curve (or not).

About the book

• Authors: George Casella and Roger L. Berger.
• Edition: 2nd edition (2002), the one these notes follow.
• Level: first-year graduate. It’s the standard theory text in a lot of statistics master’s and PhD programs, so don’t feel bad if it takes you a while.
• What you need: calculus (including multivariable integrals and series) and some comfort reading proofs.
• What it’s not: a book about software, computing or machine learning.

The big picture

The whole book keeps coming back to one idea: we have data that come from a distribution with an unknown parameter \(\theta\), and we want to learn something about \(\theta\).

Probability (1–4) → from population to sample (5) → reducing the data (6) → estimating (7) → testing (8) → intervals (9) → large samples (10) → models (11–12)

• Chapters 1–4 cover probability, random variables and the common distributions.
• Chapter 5 covers the sample mean, the CLT, and the t and F distributions.
• Chapter 6 asks which part of the data is the most important.
• Chapters 7–9 are the heart of the book, the three ways of answering a question about \(\theta\): give one number (point estimation), make a yes/no decision (hypothesis testing), or give a range (interval estimation).
• Chapter 10 shows what we can still say when exact answers are impossible but \(n\) is large.
• Chapters 11–12 apply everything to ANOVA and regression models.

Five ideas that run through the whole book

If you only keep five things from the book, keep these:

1. The likelihood is the center of everything. The MLE, likelihood ratio tests, the Fisher information and the Bayesian posterior are all built on it. (Chapters 6–10)
2. You can compress the data without losing anything. A sufficient statistic keeps all the information about \(\theta\), and the best estimators and tests only depend on it. (Chapters 6–8)
3. Unbiased isn’t always better. Estimators trade bias against variance, and what matters is the total error. (Chapter 7)
4. A 95% confidence interval is a property of the method, not of your interval. The 95% is how often the procedure captures \(\theta\). The Bayesian credible interval is the one that talks about \(\theta\) given your data. (Chapter 9)
5. With enough data, almost everything is normal. The CLT, Slutsky and the delta method are basically why z-scores, Wald intervals and chi-squared tests work in practice. (Chapters 5 and 10)

The series

1. Part 1: Foundations (chapters 1–4). Sets and counting, probability, the common distributions, and multiple random variables.
2. Part 2: From Sample to Estimate (chapters 5–7, coming soon). Random samples and the CLT, sufficiency, and how to find and evaluate estimators.
3. Part 3: Testing and Intervals (chapters 8–9, coming soon). Hypothesis tests, p-values and confidence intervals.
4. Part 4: Large Samples and Models (chapters 10–12, coming soon). Asymptotics, the bootstrap, ANOVA, and linear, logistic and robust regression.

When the four parts are out, the summary will also be on a single page, for when you (or I) just need to look something up.

How to read it

It’s something like half reference sheet, half guide. Every chapter:

• opens with the main question it answers,
• gives the key definitions and formulas, each one with a few lines on what it actually means,
• and ends with what it builds on, where it’s used later, and the link to the next chapter.

You can use it on its own to refresh a topic, or read it alongside the book. The notation is explained at the start of Part 1. Please leave some feedback in the comments so I can improve the way I draft these summaries.

Miguel