Whether or not to take real analysis

[kidbayes]

After taking Analysis I at my university, I am currently considering whether or not to take Analysis II. Since we are on the quarter system (each course lasts 10 weeks), Analysis I only covers metric spaces, topology, and continuity, while Analysis II and III cover differentiation and integration, respectively. The reasons I am considering not taking it are because the professors teaching it this quarter are not the best, and according to their course reviews, their courses are harder than usual, and I have also heard it is also very important to get an A/A- in real analysis when applying to PhD programs in statistics. However, I only have one quarter left until graduating in June. I was able to an A- in Analysis I, and I have also taken a very watered-down semester-long (16-week) course in analysis, although at a small university that most programs I would apply to would have never heard of. In my honest opinion, if I were to start a PhD program in statistics today, I believe I would need to take some real analysis course(s) in order to be fully prepared (my research interest is machine learning, which I believe relies heavily on it).

My question is for admissions to PhD programs in statistics, how much would it hurt my application to only take Analysis I and not II or III? I would like to be competitive for top programs, if possible.

Any insight or advice would be appreciated - thanks

Edited March 16, 2023 by kidbayes
clarification

[dirichletprior]

As someone who applied this cycle, I would encourage you to take as much math (specifically analysis) as possible.

I think that, in general, applicants to Statistics PhD programs are coming in with more and more math background every year. Since the GRE/Math GRE are being phased out of admissions, the only way the committee can assess your math background is via your previous coursework. In fact, one of the professors on the admissions committee of a school to which I was admitted told me that every year, they are looking to admit students with stronger and stronger math backgrounds. 

And admissions aside, taking more analysis will ease the transition from undergrad to PhD.

[bayessays]

I don't think you can go wrong with the advice to "take more math", but I also don't think you can go wrong with the advice "make sure you don't do poorly in math classes."  Taking this class will, in all likelihood, not make or break your PhD admissions but getting a B or C will hurt.

Since you already have some analysis background, "doing analysis" as it relates to stats/ML is essentially using the ideas of convergence etc and applying them to probability/algorithms problems - you'll have to learn these things in the context of statistics anyways and you'd be better off just studying them in context.  For example, you'll take stats classes and learn things about convergence in probability -- you'd be better off learning about convergence in probability directly than taking a bunch of real analysis classes in "preparation" to do statistics stuff.  (of course, there are exceptions and if you want to do extremely theoretical research and go to UChicago or something, you probably want to take more math, but if that were the case, I think you probably wouldn't be asking this question)

[kidbayes]

6 hours ago, bayessays said:

I don't think you can go wrong with the advice to "take more math", but I also don't think you can go wrong with the advice "make sure you don't do poorly in math classes."  Taking this class will, in all likelihood, not make or break your PhD admissions but getting a B or C will hurt.

This is similar to how I feel about these courses. My record has been fairly good thus far, and I feel that I have more to lose at this point (10 weeks until graduation) by taking analysis II and getting a B than taking some other math course (e.g., Graduate PDEs) and getting a B or not taking a second course at all.

6 hours ago, bayessays said:

(of course, there are exceptions and if you want to do extremely theoretical research and go to UChicago or something, you probably want to take more math, but if that were the case, I think you probably wouldn't be asking this question)

I actually am at a place like UChicago and my research thus far has been very highly theoretical, but at this point I think I'm more interested in a PhD for better employment opportunities in a field like biostatistics (although the research I'm most interested in is in machine learning theory)

Edited March 18, 2023 by kidbayes