Egnargal

Yes, exactly. I am not sure whether I agree fully, but The Bayesian Choice was described to me as a book that reads like a captivating adventure novel and that makes for good, relaxing nightly reading. BDA3 is very good, and now that the chapters and related code are available online, it is even more useful. The first Bayesian book I read was Hoff's A First Course in Bayesian Statistical Methods, which I would still recommend to someone new to Bayesian statistics. There are also nice works, like Robert and Casella's Monte Carlo Statistical Methods or West and Harrison's Bayesian Forecasting and Dynamic Models, which cover more specific areas.

These are all great replies of course and following the advice given in them will certainly benefit you. For me, reading Resnick's A Probability Path was helpful, but now after doing two semesters of probability, I prefer Durrett's or Billingsley's treatment of the material. I would also mention that once the program begins, you will likely be consumed by all the coursework, and I found it rewarding to look at some topics that I was interested in but that wouldn't be immediately relevant, knowing that I probably wouldn't have as much time to devote to them once the program began. For me, I took the time to learn more about combinatorics and topology, in particular. (Of course, such topics could become relevant in the context of research, but I investigated them more for my own enjoyment at that point than out of any desire to get ahead of the material.) As for Bayesian statistics itself, I am partial to Robert's The Bayesian Choice, which is written in an engaging style and covers a fair amount of ground in establishing the Bayesian paradigm.

I think I have seen some similar posts on here before concerning whether to accept such an offer or try to reapply next year after doing additional coursework, research, etc. But I guess it comes down to preference and timing. I was accepted into some top-ranked program but ultimately chose to attend a lower-ranked one owing to the research fit, the desirable location, the offer quality, and the program structure. Going in, I had not expected to end up where I did, so I would not discount a program just because of its ranking. It seems that you have a strong profile, and, as was mentioned, this year might have been more competitive than others, but by waiting and reapplying, you still run the risk of not being admitted again, at least not to the very top programs. I am sure that it will be beneficial to take analysis, and knowing more linear algebra is always helpful, but I would still consider whether this alone will be enough to tip the scales in your favor at the top programs. But you know best your situation and your ambitions. Even at the top programs, securing a tenure-track position after graduation depends also on what you are working on and the connections you make, so while top programs might offer better access to these things, I don't think choosing a lower-ranked program precludes a successful academic career, especially if you go in with the intention of going into academia after graduation.

We seem to have somewhat similar backgrounds: I took abstract algebra first and then was finishing real analysis and topology when PhD applications were due. No programs asked for the fall grades of those courses--except after acceptance of course to clear the degree--and I was accepted with fellowships to all the programs to which I applied, including a few top programs. While there are other factors to consider, I would think that since you hold a master's degree in applied statistics, and assuming that you performed well in the other graduate math courses you have taken, such as abstract algebra, you should be fine applying concurrently with taking analysis. (For reference, I came from a non-quantitative undergrad. My interest, however, is mainly in theory, which tends to require more advanced mathematics.)

In determining which school I would attend for a PhD in statistics, I was especially attentive to the number of courses required to complete the program. Today, I happened to look at PSU's PhD stats program, and the number of required core and elective courses seems excessive! In comparison, the program I am in now has at most one year of required courses. For anyone in a PhD program that requires a large number of courses, how was your experience? Did you find the courses to be valuable ultimately? How did you manage to complete the courses and get started on research?