Coming from a rigorous background in analysis and modern probability theory, I find Bayesian statistics straightforward and easy to understand, and frequentist statistics incredibly confusing and unintuitive. It seems that frequentists are really doing bayesian statistics, except with "secret priors" that aren't well motivated or carefully defined.
On the other hand, a lot of great statisticians who understand both perspectives ascribe to the frequentist perspective, so there must be something there that I just don't understand. Rather than giving up and declaring myself a Bayesian, I'd like to learn more about the frequentist perspective to try to really "grok" it.
What are some good references for learning frequentist statistics from a rigorous perspective? Ideally I'm looking for definition-theorem-proof type books, or perhaps hard problem sets that, by solving them, I would gain the right mindset. I've read a lot of the more "philosophical stuff" one might find searching the internet - wiki pages, random pdfs from .edu/~randomprof sites, etc - and it hasn't helped.