Mathematical Statistics: Exercises and Solutions by Jun Shao (20051223)
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Review Text
This book has all the ingredients of what in my opinion constitutes an excellent mathematics text: rigorous, concise,selfcontained, clear, and taking an abstract point of view. Note however that, due to the latter ingredient, the author studies statistics using a measuretheoretic 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 graduatelevel 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 calculusbased and makes some simplifying assumptions (e.g. continuous or discrete) about the distributions, which helps make the math digest easier.
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.
The book that is available in the kindle digital version isn't the same book as the one available in hard cover... why would amazon list them on the same page as "different options" when they are clearly not the same product?I am thoroughly disappointed about this, and I can't even find a way to ask for order cancellation or a refund :(
Too many seemingly unrelated topics with little attempt at unification. Equations rather than concepts. No attempt to relate concepts to practical applications. Nothing on the L1 norm. The brief section on permutation tests (and permutation tests applied to ranks) is laughable.On the plus side, contains many exercises of value to those whose emphasis is on distributions. Good support for the use of the bootstrap in hypothesis testing though no attempt is made to unify or select among the varying approaches to improved CI's.
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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!