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It's probably not as rigourous as you mention in your edit but I'll leave it here in case someone else is drawn to the title.
How I learned: Early in my career, I was around DC doing mostly work in applied math and computing for US national security. No joke -- constantly the work was heavily probability, statistics, and stochastic processes. I had a good ugrad math major but no courses in any of those three subjects. So I was thrown into the deep end of the pool and was constantly struggling to understand. I did pick up a good overview and…
E[(Y - f(X))]^2
of course, and as later in the derivation, should read
E[(Y - f(X))^2]
I'll have to advise my typist to do better in the future!!!
These are two very different approaches. Mathematical statistics goes trough measure theoretic probability, it connects statistics and mathematic and unifies the discrete and the continuous cases into a coherent whole.
Examples of both
- All of Statistics: A Concise Course in Statistical Inference by Wasserman
- Mathematical Methods of Statistics by Cramer
Statistics by Freedman et al. All exercises use real data as cases, I learned a lot more from them! https://www.amazon.com/Statistics-4th-David-Freedman/dp/0393...
Statistics by Freedman et al. All exercises use real data as cases, I learned a lot more from them! https://www.amazon.com/Statistics-4th-David-Freedman/dp/0393...
The sequel to this book -- Statistical Models: Theory and Practice -- and it's accompanying (well, this is how I was encouraged to read them) text -- Statistical Models and Causal Inference: A Dialogue with Social Sciences -- are excellent for intuition building and accumulating a number of examples for personal reference.
Statistics already had a lot of useful random trivia and left me with an impression that this is how a good student's resource should look like (my university had a really bad stats course so I had to learn stats myself).
But now you say there is more? Thank you random internet stranger!
Statistics has four major branches: inference, exploratory data analysis, experimental design, and visualization.
Your question is probably mostly asking about inference. I like the first few chapters of Kiefer's book to get the context of decision theory. Then you're going to have to wander far and wide, but there have been some good recommendations. Since you're digging into Dedekind on the side, you might like to go back to some of the classics here as well: Wald's "Statistical Decision Functions" and Savage's "Foundations of Statistics."
For exploratory data analysis, Tukey's book "Exploratory Data Analysis" remains the place to start, if you can find a copy that doesn't cost a fortune. Casella wrote a book on experimental design that's solid. For visualizations Wilkinson's "Grammar of Graphics" and Tufte's books are the usual recommendations.