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on September 30, 2002
This book has all the ingredients of what in my opinion constitutes an excellent mathematics text: rigorous, concise,
self-contained, clear, and taking an abstract point of view. Note however that, due to the latter ingredient, the author studies statistics using a measure-theoretic approach; and thus I highly recommend that a potential reader first study measure theory as a prerequisite. The first chapter reviews the basics of measure theory, but it may seem too giant a first step for some readers.
The first two chapters of the book give a nice overview of probability and statistics, while the remaining chapters expand on three fundamental areas of statistical inference: estimation (both parametric and nonparametric), hypothesis tests, and confidence sets). And I must admit that I'm very impressed with the author! For if a textbook is a reflection of what an author knows about some subject, then Shao represents a treasure trove of knowledge that is so eloquently shared in this book. Anyone serious about doing graduate-level reasearch in statistics should invest a year of studying this book. But be forwarned that most likely one will find this, due to the onslaught of measure theoretic analysis, one of the more challenging books to makes its way on the book shelf. For those who cannot stomach so much analysis, but would like to at least understand the gist of statistics, I recommend Roussas's book of the same title. It is calculus-based and makes some simplifying assumptions (e.g. continuous or discrete) about the distributions, which helps make the math digest easier.
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on March 2, 2006
I don't know if statistics are just that difficult a subject or statistics writers just aren't good. Either way I have not found a satisfactory statistics book that treats the subject rigorously, but still readable. This book is an excellent reference. However, it's notation is cumbersome, if you're not used to it.

Before I started taking the class that uses this book, I took four undergraduate probability and statistics classes, as well as studied advanced topics such as measure theory. That said many of the things in statistics I thought I understood, I found out that I do not, or had a hard time translating my undergraduate knowledge to this level. As with many advanced math subjects, the definitions are not enlightening and no motivation or further discussion is given for most definitions. These definitions are designed to fit into as general theory as possible, but trying to understand why some things are defined the way they were, and what the original intent of the object was, is just not there.

To use this book, you will definitely need the guidance of an expert statistician.
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on January 12, 2004
Don't waste your time: this is a rigorous book on mathematical statistics, done right, for mathematically mature readers.
If you want a plug and chug manual, buy something else. If you want precision and rigor, buy this.
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on December 7, 2003
Very readible, precise and concise treatment of statistics. Requires mathematical maturity. Although it doesn't require a background in measure theory, some familiarity (or willingness to learn) would be really helpful (Ch. 1 provides an overview of measure-theoretic probability). I read the first half of it in a PhD level statics class. I found its approach refreshing after taking an engineering oriented senior level/grad statistics class. I still frequently consult it.
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on October 7, 2004
Update: In 2010 I am using this book again to review probability and statistics in preparation for applying to a PhD program in Finance or Risk Management. I still find the book to be extremely clear. Everything still seems up to date, surprisingly. I like that fact this book has a longer useful life than an iPod or a cell phone, but is a LOT cheaper :-). The sections are very concise, sometimes just 2-3 pages, so this is definitely a reference book and not a learning book. I think it's most useful for a quick comparison of the different methods for someone who has already learned most of the material before.
I know it must be a sign of extreme geekyness to be reviewing statistics books... but it happens to be one of my passions. (So that proves it takes all kinds of people to make the world go around.) I find this book to be unusually clear. Printing is also of high quality and I did not spot sloppy notation errors. I would judge the level to be about first or second year PhD level. First chapter lays out probability theory very well and introduces the more standard notations. I find books that use the less standard notations to be annoying. I got this book to use as a reference book rather than as a textbook. I wanted to have a concise place to look up and compare the different methods. If you are learning this material for the first time, I strongly suggest you take at least one applied statistics course first. I don't think one can learn statistics easily without using data and actually running the models. Also this is definitly a graduate level book. I don't think it will be a good idea to try it before reading through several undergradute-level books on probability, regression, and statistics.

For more descriptive graduate-level books in econometric, "The Practice of Econometrics" by Ernst R. Berndt is good for hands on practice. Kennedy's "A Guide to Econometrics" provides a descriptive explanation of the various models.
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on January 25, 2015
Having read through all chapters in this book extensively, I can say that it is very poor both as an instructional material for higher level math stat and also poor as a reference. While many reviews consider this to be a "rigorous" and "precise" textbook, it has many flaws and is mostly direct copy/paste from two other better, more comprehensive textbooks. Shao takes the theorems from the classics Theory of Point Estimation by Lehmann and Testing Statistical Hypotheses by Lehmann/Romano and strips them of all context, intuition, and usefulness. This is basically a dictionary of theorems and propositions. Additionally it has one horrifyingly incomplete chapter on probability (mostly taken from Billingsley). Avoid this book at all costs and go for the original sources (and for a more advanced and comprehensive coverage of asymptotics see Asymptotic Statistics by van der Vaart, which is beautiful in its comprehensiveness, rigor, and lucid explanations)
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on November 29, 2012
I bought this book as a "capstone" book for my personal studies in statistics. I have undergraduate and masters degrees in applied mathematics (operations research with minor in environmental engineering), but my studies were broad enough that I did not get to an advanced level of statistical understanding. Once I started working, I found that while my other math skills are sometimes useful (optimization, simulation, analysis) it was statistics that is the most frequently called upon by others in my work. However, I realized that although I knew a lot of statistics at the applied (and slightly beyond) level, I felt that I couldn't "see the forest for the trees", as I did not know the advanced theories that connected all the various statistical methods.

If you've also felt that you have lots of "trees" but cannot see the "forest" that is statistics, Jun Shao's book will solve this! He keeps his basic topic list short and focused, and shows how they underlie essentially all of statistics. Also, the first chapter on probabililty theory is also very good. I never took measure theory (nor real analysis for that matter), but his expanations and presentation are clear enough that you can pick it up pretty well. However, if you are in the same boat as me, you will need to make sure you work though his examples yourself to be sure you could repoduce what he did and you know why he got that answer -- without this check, you may merely feel that you understand it, but will be missing the subtle points in the theorems -- remember: in advanced math, if its not forbidden in a defintion, it's allowed.

From there, his other chapters build nicely on one another. Starting with basic problems/motivation and fundamentals and ending at the three pillars (some would probably say raison d'etre) of statistical inference, he builds a very solid conceptual house! I have come away from this book with a much deeper appreciation and understanding of statistics. It took a while (some reviewers have cited about a year to get through this...which was correct for me!) but I am now not just more conversant in statistics, but also much more creative and flexibile in how I apply statstics, and evaluate and develop new methodologies.

Thanks Dr. Shao for creating this comprehensive and clear text!
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on January 17, 2015
Does anyone receive this book as a corrected fourth printing as described? I received one that is not the corrected printing every time.
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on October 30, 2002
This is a pretty technical book on theoretical statistics based on measure-theory. It's very well written. For Ph.D. students or readers with experience in analysis/measure-theory, it's a good investment. For less technical book, I would recommend Casella and Berger's Statistical Inference.
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on September 19, 2013
This is for PhDs in Mathematical Statistics. Not for any people from applied area. So if you are not trained to be mathematician, don't touch this book. You need advanced calculus, linear algebra, real analysis, mathematical probability knowledge to even start the first page. The author talked about statistical problems as a mathematician. Don't be confident if all your statistical training is from applied area.
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