Although not particularly advanced, this book is quite specialized and in my opinion, too narrowly focused for a book at this level. It is not a comprehensive introduction to statistical inference. Also, it often focuses on the "what" and the "how" while ignoring the "why".
The book's strengths are self-evident. The exposition of probability theory is excellent, and presented with an eye towards its use in statistics. The mathematical aspects of this book are clean and thorough, and the omissions of certain difficult proofs enhance rather than detract from the book's quality. But as one progresses further in this text, there are many shortcomings. The order in which topics are presented doesn't always seem natural to me.
My main criticism of this book is that it presents a narrow view of what statistics is, and as such I think it is misnamed; "Statistical Inference" encompasses much more than what this book covers. This book is really about "classical" statistics and it does not acknowledge or integrate more modern ways of looking at things, even when they could be presented at an elementary level. The Bayesian paradigm is hardly mentioned, non-parametric approaches are hardly mentioned, and decision theory is ignored. As such, I don't see how it offers any improvement over older texts, such as Hogg and Craig.
My second criticism of this book is that it is divorced from applications; there is almost no data presented in the text or problems. Discussion of modeling is almost completely absent, and the material on distributions in chapter 3 doesn't probe very far into the particular reasons why certain distributions arise in certain situations. This remark leads into my next criticism: the book emphasizes symbolic manipulations and ignores the deeper meaning of the mathematics. I think that an understanding of the meaning is critical if one is to find useful applications of the material.
This book is clearly more suited to certain learning styles than others. People who find manipulations of equations and formulas natural will find the proofs natural and the exercises helpful. But people interested in the ideas behind the equations will find this book lacking. The proofs are clean and easy to follow but many give little insight into the meaning of the theorems. While the motivated reader can find meaning (sometimes with considerable effort), this book's approach isn't particularly pedagogical. The exercises are numerous and challenging, but the challenge is technical rather than deep--most exercises require a clever or lucky manipulation, and occasionally drawn-out calculations, and as other reviewers have pointed out, the authors do not do a good job of creating a gradient of problems of different difficulty levels. Many of the problems in advanced chapters can be solved mechanically (even though they are not easy) without really understanding the implications and meaning of the results. A few of the problems in advanced chapters require truly tedious and lengthy calculations that, in my opinion, are a total waste of a students' time.
I understand why people use this text as a textbook, but in my opinion it needs to be supplemented by something else, either by teacher who focuses on the "why" and the deeper meaning, or, preferably, by other books that do so. This book will advance a students' understanding of certain topics but it will do little to help the students connect that knowledge with applications or other related theoretical areas. Instructors should be cautious when assigning exercises from this book--there are many excellent exercises but the level of difficulty (as well as the amount students can learn from a given exercise) is highly inconsistent. In many ways, I think this book is supplemented or complemented by the text by A.H. Welsh, a book whose weak points are more than covered by this Casella & Berger text. Another book that is a better alternative is "All of Statistics" by Larry Wasserman; his book is less thorough, but more balanced in terms of perspective, and more focused on helping the reader to learn and understand the underlying ideas. As a more advanced and more philosophical text, and to cover decision theory and Bayesian methods in more depth, I would recommend "Statistical Decision Theory and Bayesian Analysis" by J.O. Berger.