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Recommendations for things to review before starting Stats PhD

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Hello all,

I've recently accepted an offer of admission to a stats PhD program, and now I'm trying to plan my summer. It's looking like due to the pandemic, I'll be spending a lot of time at my parents' house. Although I'm sure I'll spend a lot of time playing video games and reading for fun, I also want to review some math/stats topics. I know I need to review Real Analysis since I've never taken a formal class in it. My plan is to go through as much of Rudin's Principles of Mathematical Analysis as possible, and to review my notes from my undergrad statistics courses. I would welcome any and all other suggestions for how to set myself up for success this fall.

A bit about myself: I studied statistics in undergrad, and am currently spending a year working between undergrad and the PhD program, but my job has moved to part-time and may be terminated soon.

Thanks!

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Linear algebra is the big thing. These are the topics on the Linear Algebra bootcamp for UNC biostat:

1. Vector Spaces
2. Subspaces
3. Matrices
4. Matrix Properties
5. Matrix Decompositions
6. Projections
7. Vector/Matrix Differentiation

Also this is definitely opinionated but I don't think Rudin is that great for self study, obviously YMMV but we used it in my Real Analysis class this year and without the help of my prof. some of the proofs are rather...obtuse and the intuition is not super clear at times.

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Posted (edited)

I think reviewing stuff from Calculus and linear algebra would be more beneficial, personally. Your review of Calculus need not be extensive (so you don't need to review stuff like washer and disk methods or any derivatives/integrals involving trigonometry, but you should definitely be comfortable doing stuff like chain rule, product and quotient rule for derivatives, u-substitution, integration by parts, change of variables, partial derivatives, etc.). For linear algebra, it may be helpful to review some things like linear independence/dependence, vector space, rank-nullity theorem, trace, determinant, eigenvalues, properties of symmetric positive-definite matrices, and projections onto subspaces.

It may also help to review a bit of undergraduate-level probability and statistics, so you are already basically familiar with things like pdf, cdf, and certain probability distributions.   

Does the Statistics PhD program have a course on real analysis in their first-year curriculum? If so, I wouldn't bother spending a whole lot of time on this, though it may be beneficial to review it at a high level if you haven't taken a formal class in it.

Edited by Stat Postdoc Soon Faculty

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Second to everything postdoc said above. If you're looking for a probability review, the Harvard stat 110 course online is awesome and will make the first semester Casella/Berger class much easier. If you don't have measure theory in your first semester (most programs), the analysis review won't be of much use.

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11 minutes ago, bayessays said:

Second to everything postdoc said above. If you're looking for a probability review, the Harvard stat 110 course online is awesome and will make the first semester Casella/Berger class much easier. If you don't have measure theory in your first semester (most programs), the analysis review won't be of much use.

Stat110 is a class for first year college student so it is most likely very trivial for you. The instructor for 110 (Joe) has written up a book for graduate student in STAT 210 but has not published it. You can send an email to him and he will most likely share it with you.

Also I would like to comment that the real analysis is used in proofs in inference class also (not just measure theory).

A list given by Micheal Jordan in the reddit post although I personally think this is a out-dated. If you were an older person like Prof Jordan you will probably appreciate "history" of statistics more but I personally will read several of these but take time to read more modern stuff.

image.thumb.png.82be6c38745e558bc92e5a779bfdb5a2.png

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Unless you're going to Stanford, Chicago, Penn, etc, most programs are going to start with a Casella/Berger course for which you will not need analysis. If you know the material in Stat 110, you know 90% of a first semester Casella/Berger class.  Going over the books in that Jordan list for someone who doesn't already have a master's degree-level background would be crazy.  Those are literally the books you will spend three years going over in your PhD program.  

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40 minutes ago, Stat Postdoc Soon Faculty said:

Does the Statistics PhD program have a course on real analysis in their first-year curriculum? If so, I wouldn't bother spending a whole lot of time on this, though it may be beneficial to review it at a high level if you haven't taken a formal class in it.

They list both a masters-level and a phd-level measure theory course on their course listings page (the phd-level course being required for the program) but it lists an undergraduate real analysis as prerequisite for the masters-level course. I think I'll plan on doing a cursory overview of real analysis this summer and then taking the masters-level course. 

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I think real analysis is a good choice, although I think that Rudin is awful textbook to self-study from, especially for the first time. Rudin really shines if you've already learned real analysis and you want to go over it again (and I think many should do this, because mastery doesn't come from learning something once, but reviewing the material over and over again). For a first time introduction to real analysis, and you don't have a really strong math background, I think Ross is the standard textbook. For an honors level introduction to analysis I would highly recommend Pugh since it is filled with pictures that help illustrate many concepts.

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On 4/13/2020 at 4:19 PM, statsday said:

@DanielWarlock Kreyszig is pretty heavy without already haven taken a real analysis course. 

Oh yeah I had two semesters of analysis when I went through Kreszig. I met some impressive students who jumped into that with only calculus and linear, but they were very adept learners. Also I can't imagine that many stat students would need to actually take a dedicated course in functional ever. Some will, and personally it's one of my favorite books, but not a pre req for any programs I looked at (and imagine most don't).

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