I'm a first-year graduate student working on probability and I am learning statistical inference by myself, I have skimmed through a few books like Casella /Hoggs but I find they omitted lots of details, for example, they didn't introduce the conditional expectation, so there are only proofs in discrete case about "sufficient statistics " "factoring theorem ", etc. could you recommend me a book for graduates or doctor degree that cover basic ideas of statistical inference and rigorous proofs? thanks!
One book not mentioned above which I quite like is Theoretical Statistics, Topics for a Core Course by Keener. It is relatively rigorous but quite readable at the same time. Personally though, I think a subset of Theory of Point Estimation by Lehmann, Mathematical Statistics by Shao, and Keener should cover almost all the topics at the level you want.
Take a look at Testing Statistical Hypotheses by Erich Lehmann and Joseph Romano. I also like Statistical Inference by Casella and Berger.
I recommend the book Mathematical Statistics by Keith Knight, a very friendly introduction that covers all the basics you need to know to study statistics at a high level.