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I'd like to brush up on my statistics, but most books are either overly "colourful" or just plain shallow, and certainly far from a Bourbaki-esque style of exposition. Is there any "Graduate Texts in Mathematics"-like book on Statistical Inference that's both good for learning-reviewing and as a reference book?

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  • $\begingroup$ Have you looked at Wasserman's All of Statistics? $\endgroup$
    – Ian
    Jul 14, 2016 at 0:27

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Asymptotic Theory by A.W van der Vaart would be a good choice.

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Casella-Berger is a modern and rigorous book on statistical inference, widely used in first-year courses of statistics PhD programs.

Mood-Graybill-Boes is an excellent rigorous book from a few years ago, possibly now available free in PDF format. Bain-Englehardt is somewhat similar and a little easier, but still rigorous.

These are a few of many possibilities, with which I am reasonably familiar. (I am not familiar with van der Vaart's book, but @A.E.'s Answer if far from the only recommendation of it I have seen.)

However, I must say that you are not likely to find a book on statistical inference that is written in anything like the style of Bourbaki (at least not of the set theory and abstract algebra Bourbaki books I used a few decades years ago).

The field of statistics continues to get much of its inspiration and orientation for new developments from real-world problems. I think you will be happier and more successful with your study of statistical theory if you are somewhat open to the idea that much of the material you will be studying is destined for applications. Inevitably, that may lead to some of the "color" you mention. (I say this because my impression is that the Bourbaki books do not display a lot of interest in applications--even though some of the material has proved to of great use in applications.)

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