# Statistics, for the mathematically rigorous

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.

• Maybe Introduction to Mathematical Statistics by Robert Vincent Hogg and Allen Thornton Craig [Joseph McKean added beginning with 2005 6th edition] (1959, 1965, 1970, 1978, 1995, 2005, 2013, 2019). The 1978 4th edition was used for a popular 2-semester "elective" course by math graduate students when I was at Indiana University (1982-84), although I didn't take it myself (I did buy a copy of the text however), and it seems to be fairly widely used and well known. Only Riemann integration is used, no measure theory or Lebesgue integration is needed. – Dave L Renfro Dec 11 '20 at 19:18
• You might see if Statistical Inference by Casella and Berger meets your needs. – Steve Dec 11 '20 at 20:41
• See stats.stackexchange.com/questions/56385/… (and further links there) and mathoverflow.net/questions/31655/statistics-for-mathematicians – kjetil b halvorsen Dec 23 '20 at 19:10

## 4 Answers

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." • How rigorous is it? Does it correctly and precisely delineate the meaning of the various hypothesis tests? For example, that rejecting the null hypothesis at 95% confidence level does not imply that one should have 95% confidence in the non-null hypothesis? – user21820 Dec 23 '20 at 14:41
• "This chapter considers the problem of hypothesis testing. Starting from first principles, it develops the Neyman-Pearson framework, and discusses different forms of hypothesis pairs. It then uses the question of optimality of tests as a means for motivating specific test functions. This is done in a general setup for simple-vs-simple hypotheses, and in an exponential family context for one-sided alternatives. The likelihood ratio method and Wald’s method are then introduced, ... The chapter concludes with the introduction of p-values,.. and their relation to the Neyman-Pearson paradigm." – Joseph O'Rourke Dec 24 '20 at 0:49
• I don't doubt the mathematics. What I was asking about was the use of the stated mathematical theorems applied to real-world statistical analysis. Especially things like I brought up in the (second) question in my above comment. (Related cartoon: green jelly beans.) – user21820 Dec 24 '20 at 3:20
• If you don't believe me that there is a significant issue to be addressed here, I would like to hear your own attempt at a precise statement of what can be deduced from a 1-tailed hypothesis test that has been actually performed on some sample data resulting in a p-value less than 0.05, and we can go from there. =) – user21820 Dec 24 '20 at 3:24
• @user21820 These concerns don't have anything to do with mathematical, but with statistical rigor. (Your particular concern is, to be clear, addressed in any halfway-decent statistics book, not that many students remember this.) If that's the kind of issue you have then you should avoid a book written for mathematicians, which is likely to have virtually no material on connecting the mathematics to real-world statistical analysis. – Kevin Arlin Jan 11 at 14:47

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.

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 https://www.statlect.com/.

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).