# Standard reference for classical mathematical statistics?

Can anyone recommend some books that are considered to be standard references for classical (frequentist) statistics? IE, fairly comprehensive, and also, been around for a while so that typos and mistakes in formulas had a chance to be checked and corrected

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CW perhaps?.... – user28 Sep 6 '10 at 23:34
see also question on mathoverflow regarding books on mathematical statistics mathoverflow.net/questions/31655/statistics-for-mathematicians – Jeromy Anglim Sep 7 '10 at 3:18
You could specify whether you need an introduction on applied statistics, or one on (theoretical) statistical inference. I.e., do you want the framework of testing, regression and ANOVA explained or do you want to know what the central limit theorem and the inequality of Chebiyshev have to do with the weak law of large numbers? – Joris Meys Sep 7 '10 at 15:11
see also question stats.stackexchange.com/questions/414/… – robin girard Sep 7 '10 at 17:29
Joris: well, internet is already pretty good for explanations, my motivation is having something to check against when I need a statistics related formula. For instance, recently I needed a formula for P(X=x|v'x=a) where X is multivariate gaussian and v is some vector, and none of my statistics books had it – Yaroslav Bulatov Sep 7 '10 at 21:57

E. L. Lehman, Theory of Point Estimation, 1983, and its companion book, Testing Statistical Hypotheses.

(NB: The latest edition of TPE, coauthored with George Casella, has not been getting good reviews on Amazon, but the original is still a classic.)

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I have found Statistical Inference by Casella and Berger to be a relatively comprehensive introduction.

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I'd recommend Theory of Statistics by Mark Schervish.

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 +1: i too learned from Lehman (very good reference), but this one is not mentionned nearly enough – user603 Sep 7 '10 at 16:05
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I don't think that's been around for a while, and it's not very comprehensive either...one reason I started this post was because I found an error there, and also couldn't find some things I needed, like density formula for functions of random vectors – Yaroslav Bulatov Sep 7 '10 at 3:30
It may be useful if you email the author. Perhaps, it is a misunderstanding and if not it will help in correcting the error in the next edition. – user28 Sep 7 '10 at 13:53
I have confirmation from author, so it'll probably be corrected in next edition – Yaroslav Bulatov Sep 7 '10 at 16:40
Could you, with the permission of the author, post the error briefly as a comment here? That will be useful to everyone. – user28 Sep 7 '10 at 17:04
Theorem 14.6, Sigma is singular so density isn't defined, this is fixed by instead considering distribution of $\hat{p_i}$ which is $\hat{p}$ with $i$'th component dropped (covariance matrix will be $\Sigma$ with i'th row/column removed) – Yaroslav Bulatov Sep 7 '10 at 21:50

A comprehensive and authoratative reference is Kendall's Advanced Theory of Statistics

• Volume 1 Distribution Theory

• Volume 2A Classical Inference and Linear Models

There is also a Volume 2B but it is Bayesian Inference.

Other than those, I agree the Casella and Berger is an excellent reference at the graduate level, and suggest Bain and Engelhardt's Introduction to Probability and Mathematical Statistics for upper-level undergraduates.

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 Does the remark "There is also a Volume 2B but it is Bayesian Inference." mean that Volume 2B should not be included in the description "comprehensive and authoratative (sic) reference"? because it does not have one or both of these properties, or it should not be included because it deals with Bayesian Inference rather than with purely frequentist approaches as Volumes 1 and 2A presumably do? – Dilip Sarwate Jan 15 '12 at 13:38 Hey, don't get so defensive. The OP specifically asked for Frequentist. He was just staying on topic. – Shea Parkes Jan 15 '12 at 13:47 @Shea, Dilip's query doesn't strike me as defensive. It sounds like a request for clarification. It seems unlikely to receive an answer though, since the answerer hasn't visited the site in several months. – cardinal Jan 15 '12 at 18:47