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What is a good introduction to statistics for a mathematician who is already well-versed in probability? I have two distinct motivations for asking, which may well lead to different suggestions:

  1. I'd like to better understand the statistics motivation behind many problems considered by probabilists.

  2. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.

I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source.

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As you said, it's not necessarily the case that a mathematician may want a rigorous book. Maybe the goal is to get some intuition of the concepts quickly, and then fill in the details. I recommend two books from CMU professors, both published by Springer: "All of Statistics" by Larry Wasserman is quick and informal. "Theory of Statistics" by Mark Schervish is rigorous and relatively complete. It has decision theory, finite sample, some asymptotics and sequential analysis.

Added 7/28/10: There is one additional reference that is orthogonal to the other two: very rigorous, focused on learning theory, and short. It's by Smale (Steven Smale!) and Cucker, "On the Mathematical Foundations of Learning". Not easy read, but the best crash course on the theory.

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    $\begingroup$ I've accepted this answer on the somewhat capricious basis that I now remember Wasserman's book being recommended to me by someone else several years ago. The same person also recommended "The Cartoon Guide to Statistics" by Gonick and Smith. $\endgroup$
    – Mark Meckes
    Commented Jul 28, 2010 at 19:13
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    $\begingroup$ errata for "Theory of Statistics" by Mark Schervish: stat.cmu.edu/~mark/advt/.index.html $\endgroup$
    – vasili111
    Commented Nov 11, 2015 at 12:10
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Mathematical Methods of Statistics, Harald Cramér is really great if you're coming to Statistics from the mathematical side. It's a bit dated, but still relevant for all the basic mathematical statistics.

Two other noteworthy books come to mind for inference and estimation theory:

Not entirely sure if this is what you wanted, but you can check out the reviews and see if they meet your expectations.

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I loved the Freedman, Pisani, Purves' Statistics text because it is extremely non-mathematical. As a mathematician, you will find it to be such a clear guide to the statistical concepts that you will be able to develop all the mathematical theory as an exercise: that's a rewarding thing to do. (The first edition of this text was my initiation to statistics after I completed a PhD in pure mathematics and I still enjoy re-reading it.)

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You will find many applications of Mathematical Statistics in 'Mathematical Statistics and Data Analysis' by John A. Rice. The 'Application Index' lists all applications discussed in the text.

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I think you should take a look to the similar post from mathoverflow.

My answer to this post was Asymptotic Statistics by Van der Vaart.

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For you I would suggest:

Introduction to the Mathematical and Statistical Foundations of Econometrics by Herman J. Bierens, CUP. The word "Introduction" in the title is a sick joke for most PhD econometrics students.

Markov Chain Monte Carlo by Dani Gamerman, Chapman & Hall is also concise.

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  • $\begingroup$ I guess I should take the first suggestion as a vote of confidence. $\endgroup$
    – Mark Meckes
    Commented Jul 21, 2010 at 17:52

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