Theory of Point Estimation

The order arrived in time, and has a good quality. Totally satisfied!

The primary issue with this book is that the author is trying to disseminate the knowledge in an almost prose-like form. In my opinion, this is a horrible way to teach theoretical statistics as it leaves the reader hunting and underlining left and right. In some cases, he accidentally leaves out parts of definitions. For example, when defining the sigma algebra in chapter 1; he doesn't mention that the sigma algebra must contain the empty set. While these little details might seem irrelevant in the big picture; it forces the reader to look to outside sources for definitions and doesn't serve as a compact reference.

I originally gave this book a much more generous review, but after having worked with it extensively, I have revised my review. This high-level text is aimed towards people in Ph.D. programs and theoretical researchers in statistics and related areas. The second edition of this book seems hastily thrown together, especially as one gets into the later chapters. It is difficult to use and may be misleading to many students. Its only justifiable use, in my opinion, is as a survey of the literature on point estimation up through the 90's, but even here it is not ideal. The earlier edition by Lehmann is, in my opinion, much better written.* This book has some strengths. It is rich with concrete examples. Whenever an abstract structure, framework, or idea is introduced, the authors provide numerous examples and discussion. For the mathematically mature reader, one will find that this book (unlike the more elementary Casella and Berger text) focuses primarily on deeper meaning, although chapters 4-6 seem to have more tedious formulas and manipulations than 1-3.* The problem sets have good and bad aspects: the gradient of difficulty is fairly natural and there is a diversity of difficulty levels. Many problems touch on some deeper theoretical issues and/or are useful for understanding basic concepts. However, the problems are often very poorly and imprecisely worded. Rather than making the problem sets self-contained, they often reference earlier examples and problems--sometimes one must page back to 3 or 4 different parts of the book to understand the full statement of a problem. Often, the problems require additional unstated assumptions and many are outright false as written without these assumptions. In my opinion, many of the exercises are not very useful because anyone able to understand their full and precise statement would find them almost trivial. In a sense, however, working these exercises does help one to develop the ability to dredge through sloppy work and re-state the mathematics in a precise manner, which is a useful skill.This leads into my next criticism:* The book is full of errors, especially in the exercises. Some errors are typos, but others are assertions, inconsistencies, or omissions in the discussion that can be misleading and sometimes are logical fallacies. The most common error is the omission of certain key assumptions for a theorem or exercise--without which the theorem or exercise is false. These errors can be hard to spot because the subject material is subtle and the book is filled with statements and assertions that are true but may appear at first strange or counterintuitive. This book has been in print almost 9 years as of when I am writing this review, and the publisher ought to have released at least a revised printing. I was not able to find an errata page on the internet either. Many of the errors are left-overs from the first edition that were never corrected. This is inexcusable and puts both author and publisher to shame!* For a book written at this high a level, this book is disturbingly lacking in rigor. Certain important terms (group families, invariance, Jeffreys prior, Haar measure, to name a few) are given only a vague, loose definition, and are never defined fully and rigorously. Few proofs are given, even when inclusion of the proof would greatly enhance understanding. This book is advanced enough that a student lacking mathematical maturity would not be able to get much out of it: a higher level of rigor would clarify underlying ideas rather than being a hindrance.* This book is not self-contained. While most people reading this book will own useful supplementary references, I think that the omissions in this book are too much. A brief appendix with a rigorous definition of all technical terms and a summary of certain distributions would be welcome. The 10-page index of figures, tables, and examples is, in my opinion, useless, and ought to be replaced by a useful appendix or two. The table of notation is useless since the author makes use of many notational conventions that are not included in this table.* In its use and discussion of loss functions, this book ignores the insights that decision theory has to offer. The book uses loss functions haphazardly, often on the sole basis of mathematically convenience or past convention. The book focuses almost exclusively on squared error loss and related losses, and also heavily on convex losses. I think this attitude, while appropriate to some of the problems described in the book, is misleading and sets up bad habits for students. In reality, most loss functions derived from utility theory have concave tails, and most location-parameter losses are asymmetric about 0. The book does not discuss these issues, nor does it develop any theory or machinery that could be used to tackle problems involving such loss functions. In ignoring these issues, the book loses sight of the purpose and application behind the mathematical structure. This leads into my next criticism:* This book does not provide much of a framework for relating theory to applications. Many of the examples given and referenced in the text are "toy" examples, mathematical oddities that infrequently arise in practice. In practice, much of the mathematical structure arising in statistics originates at the level of the application, and this book seems to treat the structure as if it exists on its own, for its own sake. While there's nothing wrong with a book focusing on theory, I think that the theory should be developed with an eye towards its use in a practical setting. This book's discussion of open areas of research seems skewed towards esoteric, theoretical areas that are less useful for the advancement of human knowledge than other areas which are actually more elementary and less well-developed. Perhaps this point of critique is directed not just at this book but at the field of applied mathematics and statistics as a whole. However, I think that the degree to which this book delves into "theory for the sake of theory" is rather extreme. There are so many ways theory can be developed so as to be more practically useful, and this book simply doesn't go down that path at all.* This book's treatment of Bayesian statistics is superficial and, in my opinion, the chapter on Bayesian methods is exceptionally poor compared with the rest of the book. When discussing Bayesian methods, the authors only discuss their use as they apply towards the end of obtaining "good" frequentist estimators. Bayesian thinking is not integrated into the discussion from the beginning; it is treated as an afterthought or a technique. I think this is a great loss, since a Bayesian perspective can enhance the presentation of a number of topics covered. The Bayesian chapter also has more muddled presentation, and is more technical and tedious (especially in the exercises) than the rest of the book.* Some other sections are just straight up bad. I found the sections on linear models in the chapter on equivariance to be useless because the presentation was so arcane and so distant from the original motivation that it was nearly impossible to get anything useful out of it.Bottom line? This book is widely used as a textbook, and I think that its use as a textbook is very hard to justify. If it is used as a textbook, it absolutely needs to be supplemented by additional materials--its use alone could be very confusing and also possibly misleading.As a supplement, I recommend "Statistical Decision Theory and Bayesian Analysis" by Berger. That book is a good way to "undo" the damage that this book does. In particular, it will warn people about potential ways the techniques described in this book can be misused, and it will show the student how to think properly (and critically) about loss functions. Although the name suggests a completely different focus, the books overlap more than one might expect, and that book is infinitely more well-written. It also addresses nearly all the shortcomings I mentioned above, although it certainly does not cover all of the same material. In my opinion, the material that it does not cover is both less interesting and less useful from a practical perspective.

Great book for graduate students in Statistics, Biostatistics, and probability.

The Theory of Point Estimation, 2nd Edition, by Lehmann and Casella belongs in every Statistician's library. The proofs are particularly well written in a highly accessible manner. If one wants to make clear and concise citations, this book is a must have for your thesis work.