# Book on Measure Theoretic Statistics

I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a standard course in (possibly multivariate) statistical inference.

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The closest thing I know is "Testing Statistical Hypotheses" by Lehman and Romano. –  Michael Greinecker Feb 26 '13 at 18:36
Thanks. I'll check it out. –  Evan Aad Feb 26 '13 at 19:38

I have just the right book for you. Try Theory of Statistics by Michael J. Schervish. It is the only book on measure-theoretic statistics that has received $5$ stars from every person who commented on it on Amazon.

I also suggest reading the Lehmann volumes, Theory of Point Estimation and Testing Statistical Hypotheses, which do not compromise on mathematical rigor although Erich Lehmann was a statistician.

Professor Dudley of MIT has a wonderful set of notes on measure-theoretic statistics, which I personally refer to very often. If you would like to see more, then Dennis Cox of Rice University has a set of notes entitled The Theory of Statistics and Its Applications.

Finally, let me say that it is important to read landmark papers that have introduced some of the major concepts that we see in the theory of statistics today. Having said this, I highly recommend Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics by Paul Halmos and Leonard Savage.

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Sounds great. Thanks, i'll check it out. I wonder why so few expositional works have been written on measure theoretic statistics, in particular in view of the plethora of textbooks, at all levels, that deal in measure theoretic probability. –  Evan Aad Feb 26 '13 at 19:41
You know, I asked myself the same question two years ago while I was preparing for my qualifying exam on Probability and Statistics. I was quite frustrated with all the proofs that I had seen of the Fisher-Neyman Factorization Theorem and the Neyman-Pearson Lemma, because these proofs always made simplifying assumptions in order to avoid measure theory. –  Haskell Curry Feb 26 '13 at 19:48
Dear Haskell, i've read the article you recommended (Halmos & Savage). It's beautiful and illuminating. But there's one sentence that i don't get and would appreciate your perspective. In the end of section 8, after the example that shows that pairwise sufficiency doesn't imply sufficiency, the authors write: "If, for instance, we imagine that it is important to a statistician that he either estimate $\alpha$ sharply or refrain from estimating it altogether, then he is by no means as well off with the observation of $y$ as with that of $x$". –  Evan Aad Mar 1 '13 at 19:10
It seems to me that $\alpha$ can be estimated as sharply as desired by observing $y$ alone: Simply divide $Y$ into as many equally spaced segments and note $y$'s frequency. About $1/2$ of $y$'s occurrences will be in the segment that contains $\alpha$ whereas the other segments will be visited with equal probability that is close to $0$. –  Evan Aad Mar 1 '13 at 19:11
Dear Haskell, the link you provided to Cox's course notes points to notes on probability theory rather than statistical theory. Could you please correct it? I'd really like to take a look at Prof. Cox notes. –  Evan Aad Apr 21 '13 at 18:32