Mathematical Statistics

Rating:
Summary: Great book, not for kids
Review: 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.

Rating:
Summary: Excellent, very clear, accurate notations
Review: 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 (though I was not looking intensely). I would judge the level to be about first of second year graduate 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, you may want to use an applied book at the same time. I don't think one can learn statistics easily without using data and actually running the models. Something like "The Practice of Econometrics" by Ernst R. Berndt might be helpful. Kennedy's "A Guide to Econometrics" provides a descriptive explanation of the various models that can be very helpful at times. These two books are for applications in finance or economics and may not be as good for an engineer.

Rating:
Summary: a measure-theoretic based introduction to statistics
Review: 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.

Rating:
Summary: Abstract Nonsense
Review: This book is highly abstract and technical. Very little emphasis is placed on statistical thought or ideas. It is easy to get lost in this sea of analysis and mathematical trickery. The reader is left with the impression that the author is more interested in demonstrating his mathematical prowess and ability to pointlessly abstract concepts than teach. Consequently, this book is unsuitable for teaching--unless the reader has at least a MS level mathematics education and plans to do research in mathematical statistics. Even then, this book is, at best, a good reference.

For advanced textbooks in measure theoretic mathematical statistics, I highly recommend:

Theory of Point Estimation
Lehmann and Casella

Testing Statistical Hypotheses
Lehmann

Rating:
Summary: Abstract Nonsense
Review: 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.

Rating:
Summary: Worth the time reading
Review: 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.