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im a math student who wants to study statistics in depth for a master degree(maybe doctor degree). And I was wondering what kind of math knowledge might be needed for upper level statistics. I wanna learn statistics in a mathematically rigorous way and understand all the maths behind statistics knowledge.

I have searched relevant questions on a couple platforms and the answers are linear algebra, calculus, and analysis. But what analysis exactly do I need to study? I can think of real analysis, functional analysis, measure theory and maybe a bit of topology?(not very sure if these are actually needed) Are these enough? Is there anything else I need to study in depth or is there anything I don't need to spend too much time on?

Also any math books recommended for learning statistics in a theoretical and mathematical way? I have a couple books in hand such as mathematical methods of statistics by Cramer, but its simply not for me yet. I found it a bit challenging to read(since im still doing my undergrad degree in math) and that's why im asking for some math books to read first and then get back to this book for further study.

Thanks in advance.

Jared
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    This sounds like the question answered at https://stats.stackexchange.com/questions/118112. Closely related are https://stats.stackexchange.com/questions/402333 and https://stats.stackexchange.com/questions/463248. – whuber Sep 21 '22 at 21:58
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    Something worth remembering is that, while many people will enter statistics master’s and doctoral programs with bachelor’s degrees, probably not many will already have a PhD in mathematics. Statistics programs seem to train their PhD students just fine without making them earn math PhDs first. – Dave Sep 21 '22 at 22:00
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    Unless you really pivot towards something like topological data analysis, I don't think a lot of topology is required. Some results require compactness, but I wouldn't recommend trying to master everything related to topology (e.g. Poincare-Hopf theorem) just because topology is occasionally used in statistics. Unless you specialize to something using topology, of course. – Galen Sep 21 '22 at 22:36
  • @Galen Thank you so much for your advice! – Jared Sep 21 '22 at 23:08
  • @Dave Thank you Dave – Jared Sep 21 '22 at 23:09

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As a start to study Statistics mathematically and rigorously, I would recommend you to read:

It is a very good book to begin your journey.

Átila Correia
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    thank you so much for your help. Yes I have tried to read this book before but I felt like this book was quite challenging and I could hardly go through even the first few chapters.. Do you think I should be reading some books else before diving into this one? – Jared Sep 21 '22 at 22:13
  • @Jared As a suggestion for preparation, I advise you to read some real analysis books before studying Shao's Mathematical Statistics. You could start with Terence Tao's Analysis I and Analysis II books. Once you have mastered them (or any other books which have the same content), it would be interesting to study measure theory within the context of probability. As a recommendation, I think Dudley's Real Analysis and Probability would be enough to equip you with the machinery needed to proceed your studies. – Átila Correia Sep 21 '22 at 22:26
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    nice. really appreciate your help. – Jared Sep 21 '22 at 22:48
  • after having finished books on analysis and having read Dudley's Real Analysis and Probability, do you think one should be able to read through Shao's Mathematical Statistics with ease? or at least one could manage to finish it without too much difficulty? – Jared Sep 21 '22 at 23:14
  • @Jared As far as I can tell, I think so. – Átila Correia Sep 21 '22 at 23:27