Law-of-large-numbers

Erich Lehmann is well known for his advanced statistical texts on hypothesis testing and estimation. he has also written a nice intermediate level text on nonparametric methods based on ranks. This book is another advanced text providing a thorough treatment of asymptotic (large sample theory) methods. It is very modern and includes such popular current topics as bootstrap and density estimation.

Tom Ferguson's book is the standard at the UCLA Department of Statistics and for good reason. The book follows a logical format, essentially proving a different limit theorem/approximation in each chapter. The book is good for an advanced graduate 1 quarter/semester course in asymptotic theory, although some of the topics may have to be omitted. I wouldn't recommend reading this book by yourself since I find it to be very compact/concise. However, if you've taken a similar course already it makes an invaluable reference.

It covers important topics, and it has a clear exposition

It is almost impossible not to recommend a book by Professor Ferguson, and this book is no exception. I will deviate slightly from typical book reviewers to mention a few noteworthy things common to Professor Ferguson's books. First of all, he writes mathematics clearly, concisely, logically, and in an organized manner. He is therefore an exception to the typical mathematics researcher whose writings look like running notes from a gauntlet runner or a gladiator running from a lion in an ancient Roman arena. I first learned graduate statistics from his 1966 book which I believe is titled Decision Theory or Statistical Decision Theory, and that book is as up to date in its information (aside from incorporating intervening studies) as though it were written today. Readers even outside mathematics should demand a reprint of that book if they want to learn real statistics. Professor Ferguson's character (I have met him) is as honest and open and logical as his books. His books do involve Lebesgue integration, as some other reviewers have mentioned, and I recommend that even non-statisticians hire a consultant or tutor to either teach them Lebesgue integration or to translate into approximate English or at least elementary mathematical language what Lebesgue integration does. I will try to discuss it myself either in a later addition to this book review or in another book review. My only criticism of Ferguson's books concerns the lack of representation of probabilistic alternatives to Bayesian methods (which I have been developing since 1980) in which, instead of dividing probabilities one substracts them and adds a constant. These have the advantage of being defined even when events have probability zero, unlike (Bayesian) conditional probability, and probability zero events are surprisingly common (e.g., lower dimensional events, extremely rare events assuming continuous random variables, etc.)unlike most people's impression - precisely because of arguments involving Lebesgue type integration. You can find abstracts of some of my papers on this at the Institute for Logic of the University of Vienna (on the internet).