Statistics, for the mathematically rigorous

Mathematics Educators Asked by John Clever on January 13, 2021

I don’t know where I can find a rigorous statistics course or textbook. The closest thing I can think of is measure-theoretic probability theory, but I wouldn’t really call that "statistics". By ‘close to statistics’, I mean something that, although as mathematically rigorous as probability theory, can be reasonably substituted for a statistics course with minimal study.

Does such a thing exist? Perhaps not, as the point of many statistics courses is in their applications.

4 Answers

Kevin Arlin already gave my "best" answer in All of Statistics. That said, statistics is a large topic. If you're looking specifically at inference and statistical learning and want a rigorous exposition of most of the different methodologies used I would recommend Elements of Statistical Learning, although it assumes a heavier mathematics background than you might expect (in particular, matrix differentiation and probability).

Answered by Nate Bade on January 13, 2021

As a student of mathematics taking a statistics course next semester, I have been plagued with this problem. I have found a solution in "Lectures on Probability Theory and Mathematical Statistics" by Marco Taboga, which is a collection of many lectures and exercises from the website

Answered by Kevin Osborn on January 13, 2021

I am not familiar with this book, but the title alone suggests it might be worth examining for your purposes.

Statistics for Mathematicians: A Rigorous First Course. Victor M. Panaretos. Compact Textbook in Mathematics. Birkhäuser/Springer 142 (2016). ISBN-10 : 9783319283395. Springer link.

"Intended for students of Mathematics taking their first course in Statistics, the focus is on Statistics for Mathematicians rather than on Mathematical Statistics."


Answered by Joseph O'Rourke on January 13, 2021

I would suggest "All of Statistics" by Wasserman. It is reasonably concise and moderate in its demands on background, but much more mathematically serious, also covering a much wider range of material, than a typical first course in statistics.

Answered by Kevin Arlin on January 13, 2021

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