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a measure-theoretic based introduction to statistics
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.