Book recommendation for Mathematical Stats

So I've read Tanis and Hogg, and Ross's Mathematical Statistics books and understood them fairly well. Still when I try to read some other Stats books I always encounter something I've never seen before that keeps me from making progress. To give some examples, in Gelman's Bayesian Data Analysis, I hit several things I didn't understand like the expected value formula applied to a vector. In Greene's Econometrics book I encountered several formulae for Taylor Series of many-valued functions that I hadn't see involving a single term of two Sigma expressions. In Gibbon's book on Non-parametric Stats I couldn't get much past the first chapter for not knowing some of the formulas relating to this or that distribution or the covariance of the expectation of certain things, which was never substantially emphasized in my other readings.

So anyway, I have been googling to find out if there's a better Mathematical Stats book that could prepare me for these other studies. I had Shao recommended to me, but the exposition there seemed bad (one problem dealing with Borel fields was stated so badly that it took me from a state of understanding to a state of confusion for a day). I had always heard good things about Casella and Berger but after reading a couple chapters it seems no more informative than Tanis and Hogg. Right now I'm reading Jaynes' book Probability Theory and it's going well but there's a lot of Philosophy to get through. I like it, but it's taking me a while to get to the Math.

I'm looking for a book that a) gives a very comprehensive and somewhat advanced explanation of the Math that you encounter in Stats, and b) is explained well enough for self-study. Does anyone know of such a book?
I was going to recommend you Jaynes' book but then I realised you already referred to it in your text. In grad school I couldn't find a better textbook to explain statistical theory in plain English than that one.


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If it's of any help, as an undergrad I studied from John E. Freund's Mathematical Statistics with Applications. I also studied from Jim Pitman's Probability. I like Freund's book because it seems to cover more topics. The first few sections of each chapter are theory-driven while the last section is application. I'm self studying from it now to prepare for grad school. My professor referred me to Sheldon Ross's book as well.
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