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I know this question has been asked many times but all the answers seem to suggest old books or otherwise do not seem updated. Also, most these questions do not really seem to cater to my particular interest.

I am a young PhD in mathematics (probability) looking to transition to applied statistics. I have read many "Hands-on this topic" books and have working knowledge of Python and R modules and libraries. However, I would like a reference on mathematical multivariate statistics where I would learn the theory behind many of these models. So, I am interested mostly on the theory and if the book has some "applications" that would be very nice. If I had to choose between the two, I'd prefer a reference with the theory. Some books I have read are:

  1. All of Statistics by Larry Wasserman (2005). He has a bazillion typos, and he does not really prove any of his theorems (I understand that since this is a birds eye view of all of statistics).

  2. In all likelihood by Yudi Pawitan (2001). I liked it somewhat but I felt he talks about too many properties that he does not really prove specially about the MLE which is quite bad since the book is all about MLE and likelihood.

  3. Mathematical Statistics and Data Analysis by John A. Rice (1995). I like it a lot but this is univariate only. Ideally, I would love something like this multivariate and updated to more recent developments.

  4. Introduction to Statistical Learning by James, Whittens, Hastie and Tibshirani (2013). This was an excellent intuition-only book with lots of R code. I know of its predecessor ESL but somehow I did not like that one. It pretends to be a mathematical book but it intentionally omits proofs and it often used a lot of theory that was not precisely referenced so I am looking for reference.

I also started reading:

  • Multivariate Analysis by Kardia, Kent, Bibby (1979). This is last one is almost exactly what I am looking for except this one feels old since it is already 42 years since originally published.

Also, all my references are 16 years old at the best and getting older.

If you have modern suggestions I would like to read them. If you otherwise think nothing modern fits the bill and have an older suggestion that will work, that would be fine too.

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  • $\begingroup$ Apart from the recommendations here and here, I would add Seber's Multivariate Observations. $\endgroup$
    – StubbornAtom
    Commented Jun 3, 2021 at 14:45
  • $\begingroup$ springer.com/gp/book/9780387781884 $\endgroup$
    – whuber ♦
    Commented Jun 3, 2021 at 15:20
  • $\begingroup$ @StubbornAtom I heard of Multivariate Observations. Have you read it? If so, is it rigurous and mathematical while preserving a statistical gist? $\endgroup$
    – William M.
    Commented Jun 3, 2021 at 17:13
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    $\begingroup$ I have used it in my course. Yes to both of your questions. $\endgroup$
    – StubbornAtom
    Commented Jun 3, 2021 at 17:56
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    $\begingroup$ I have seen the book and talked with Izenman (about four years ago), but I have not studied it. I thought of it immediately when I read your post. $\endgroup$
    – whuber ♦
    Commented Jun 3, 2021 at 18:34

3 Answers 3

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What first comes to mind for me is Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning (Springer Texts in Statistics) by Izenman, which has its focus entirely on newer ideas not covered at all in most more traditional books. At Amazon.com it gets very positive reviews.

I did not read it yet but ...

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  • $\begingroup$ It seems to be the right book indeed. It has the newer techniques covered in Elements of Statistical Learning but as of yet I have to know if this book includes proofs or just hand-waving like ESL. $\endgroup$
    – William M.
    Commented Jun 4, 2021 at 2:15
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Very interesting-looking is Generalized Principal Component Analysis by René Vidal. It might be more specialized than the Izenman tome, but certainly introduces new mathematics to the game!

I really want to study it, bit it will require some time to learning such things as some algebraic geometry, which was not in my studies ...

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You asked for books not regurgitating old material ... the following is not cheap, and looks more like an article collection with examples of new applications Advances in Principal Component Analysis: Research and Development but might still be interesting. An edited volume with many authors.

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