The calculus based math-stat sequence has long been a stepchild in statistical education, especially for advanced undergraduate and beginning graduate students who do not take additional courses in statistics. The sequence often consists of one course in probability theory followed by a second course in statistical theory. Many statistical educators feel that the first course contains much more probability theory than is wanted or needed for the second course; it is hard to understand how it is that being able to compute how many possible hands there are in a five card stud poker game will make anyone a better statistician. At the same time, the second course often drowns in arcane estimation methods and misses many recent developments in statistics, such as Bayesian statistics and the bootstrap method, to mention just a couple of topics.
Wasserman sets out to remedy some of the shortcomings of the standard probability-statistics sequence. He sets himself high goals, as shown in the title All of Statistics, but even he admits that the title may be a bit of an exaggeration, even though the spirit of the title is very much reflected in the content of his book. The book came about as answer to the question posed to him by computer science colleagues about where their students could turn for a quick understanding of modern statistics. Thus, the book contains chapters on graph theory, and computer science receives more of a mention than in most books on statistical theory.
Many will applaud Wasserman for his abbreviated treatment of probability theory. He only spends the first 86 pages on this topic, even though he even finds room for a chapter on convergence of random variables. Part II deals with statistical inference, and in addition to the usual topics, he provides us with a chapter on the bootstrap, and chapters on Bayesian inference and statistical decision theory. The final part of the book takes up about half of the book and deals with statistical models and methods, containing chapters on regression, multivariate models, causal inference, graphs, non-parametrics, smoothing, classification, stochastic processes, and simulation. I mention these chapter titles to show how the book contains topics not always included in courses on mathematical statistics. Finally, according to the author, students are only expected to know calculus and some algebra to use the book.
It would seem that students need a good deal of mathematical maturity beyond having taken a couple of courses in calculus in order for them to benefit fully from this book. The mathematical content of the book is denser than what is found in the typical book on mathematical statistics. I expect many students will need a good deal of help from their instructor in order for them to benefit from the terseness of the notation and the presentation of many topics. Instructors will also need to spend time on why it is that certain statements and theorems are presented they way they are and why certain topics are important. This would not be a good book for self-study! I also think that the book will not be a good choice for the text in a seminar on mathematical statistics, where the students are expected to give most of the presentations themselves.
A good review of a textbook should have inputs from students who have used the book in a course. This review does not have such contributions, and I can only speculate about what students would have said after using this book as their text. The use of the book in a one-semester course would leave most students completely breathless and wonder what had happened to them after having been exposed to all the material in the book. Using the book for two semester would make them grudgingly admit they had learned much, but maybe still too much for most of them to absorb. They would admire the elegance of the presentation and realize that they had been exposed to important matters.
Finally, I miss a sense of discovery and excitement about statistics in the book. At a randomly chosen page (299) there is a presentation of breast cancer data in a book published in 1973. May the age of the data be as it is, why are we exposed to these data? Is there anything we want to learn about breast cancer that led us to these data? What can statistics contribute that the doctors did not already know? Especially for non-statisticians, motivation for an example is particularly important. The example ends with the fitting of a model, concluding that there is no evidence that the model is a poor fit. But what does that mean for breast cancer research? Any instructor must be prepared to furnish such motivational questions throughout the course.
Gudmund R. Iversen (email@example.com) is professor emeritus of statistics at Swarthmore College. Among his interests are statistics education and uses of Bayesian statistics.