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Anyone can suggest me one or more good books on Statistical Inference (estimators, UMVU estimators, hypothesis testing, UMP test, interval estimators, ANOVA one-way and two-way...) based on rigorous probability/measure theory?

I've checked some classical books on this topic but apparently all start from scratch with an elementary probability theory.

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Many people still swear by the pair of classics by Lehman et al Theory of Point Estimation and Testing Statistical Hypotheses. If you want something a bit more modern, I like Theory of Statistics by Schervish. It covers both the classical and Bayesian theory, but does not slight either of them. There is also Mathematical Statistics by Shao, that is structured much more like the non measure theoretic textbooks, starting with a whirlwind review of probability theory, and seems to be used as a textbook fairly often judging from the semi-incoherent negative reviews on Amazon.

Probably better than my limited opinion, see the answers to a similar question on MathOverflow.

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  • $\begingroup$ These two books are at a higher level than Bickel and Doksum. They are both excellent choices, too. $\endgroup$ – ncmathsadist Jul 16 '11 at 15:14
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    $\begingroup$ He asked for books based on measure theory. Lord knows why. I am reminded of the quote of Stephen Sen "A theoretical statistician knows all about measure theory but has never seen a measurement whereas the actual use of measure theory by the applied statistician is a set of measure zero" from stat.columbia.edu/~cook/movabletype/archives/2009/05/… $\endgroup$ – deinst Jul 16 '11 at 16:15
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    $\begingroup$ I asked for books based on measure theory because we was recently required to follow a probability course based on measure theory. I suspect it is better to continue on this direction rather than restart with a statistic book based on elementary probability. Moreover I already have several book of "the other kind" and to confirm or reject my suspects I was interested to compare these books with measure theoretic books. $\endgroup$ – unlikely Jul 16 '11 at 22:03
  • $\begingroup$ @unlikely I'm not a statistican, either applied or theoretical, but I do not think that mathematical benefits that much from the more rigorous foundation, unless you like saying 'almost surely' alot. Judging from the other responses here, on MO and on stats.stackexchange.com others seem to agree. $\endgroup$ – deinst Jul 17 '11 at 16:48
  • $\begingroup$ I cannot find any reason why one should NOT study measure theory-based statistics. I think it's interesting and fun. $\endgroup$ – Taxxi Sep 18 '14 at 6:05
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Bickel and Doksum, if you can find the 1977 edition is very good. It is a high-level treatment of statistics from a mathematical (tho not measure-theoretic) standpoint.

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Dienst's recommendations above are all good,but a classic text you need to check out is S.S. Wilks' Mathematical Statistics. A complete theoretical treatment by one of the subject's founding fathers. It's out of print and quite hard to find,but if you're really interested in this subject,it's well worth hunting down.

Be sure you get the hardcover 1963 Wiley edition; there's a preliminary mimeographed Princeton lecture notes from 1944 by the same author and with the same title-it's not the same book,it's much less complete and more elementary. Make sure you get the right one!

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  • $\begingroup$ -1 : I took a glance at this book in my university library today, and whatever its virtues, it is archaic and outdated, and it does not discuss most of the topics that the OP listed in his/her question. $\endgroup$ – Adam Smith Oct 4 '11 at 4:48
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    $\begingroup$ Uh,it's also by one of the founders of the subject,Adam."Don't read the students,read the MASTERS!"- Abel. Hardy's A COURSE IN PURE MATHEMATICS and van der Waerden's MODERN ALGEBRA are both hopelessly "outdated" in terms of notation and sometimes subject matter,but both are still very strongly recommended by mathematicians.So are William Feller's AN INTRODUCTION TO PROBABILITY THEORY AND ITS APPLICATIONS and Paul Halmos' MEASURE THEORY. I still think the Wilks book is worth a look. $\endgroup$ – Mathemagician1234 Oct 4 '11 at 7:22
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    $\begingroup$ If someone asked for a book about calculus or basic real analysis and you recommended Hardy's "A Course in Pure Mathematics", I would downvote it. Similarly, if someone asked for an introduction to abstract algebra and you recommended van der Waerden, I would also downvote it. They have a certain historical value (van der Waerden more than Hardy), but they are not appropriate as textbooks. By the way, I've seen you fulminate against Bourbaki in other places, and the authors of Bourbaki are themselves masters. Why the inconsistency? $\endgroup$ – Adam Smith Oct 4 '11 at 15:24
  • $\begingroup$ @Adam There is no inconsistency.First of all,"Bourbaki" is not a single author,of course,but the mythical collective term for an organization composed of some of the great French mathematicians of the early 20th century.Secondly,the sheer level of abstraction and difficulty,given by these books,despite the modernity of the presentation,would be detrimental for a beginner.Lastly,they're simply very difficult to read,most of them. To this last comment I should add I currently don't read French.So whether this is a result of the translation or inherent in the books themselves,I can't say.(cont) $\endgroup$ – Mathemagician1234 Oct 4 '11 at 19:27
  • $\begingroup$ @Adam (cont) I happen to like older books.One of the great things about mathematics is that older books,unlike in other scientific disciplines,are not worthless because mathematics,by it's very nature,does not change in it's basic details.In fact,Tom Korner at Cambridge frequently recommends both Hardy and Whittiker and Watson's A COURSE IN MODERN ANALYSIS,which dates to the beginning of the last century,to his analysis students.You want to downvote him,too?That being said-there is a general dearth of texts on statistics at the level OP asked for.This is probably by the subject's nature. $\endgroup$ – Mathemagician1234 Oct 4 '11 at 19:33
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There's the book by Morris de Groot, and one by Bernard Lindgren. Both have bland titles that I don't remember. I think the former might be "Probability and Statistics" and the latter "Statistical Inference" or something like that.

Lindgren's book contains a proof that the location-scale family of Cauchy distributions admits no coarser sufficient statistic than the order statistic (i.e. an i.i.d. sample sorted into increasing order); maybe that's not a crucial thing but it's something you find frequently asserted but seldom proved, so it stands out in my mind.

Both books cover the topics you've mentioned, although they don't assume you've had measure theory.

Since you mention ANOVA, let me add that if you want to understand the theory, you should know things like the (finite-dimensional) spectral theorem, the singular value decomposition, etc. Many books treat ANOVA and regression without that, so you won't learn why the sampling distributions of test statistics are what they are, etc. I'm not sure which book to recommend for this right now.....

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