I want to deepen my knowledge on statistics, specifically, and probability theory in general. I want to know what textbook could be recommended for someone at my current state of knowledge looking to go deeper. Of course, for this question to make sense I should describe what my current state of knowledge is.
I have a basic but solid understanding of some of the core topics of descriptive and inferential statistics; mainly
- The normal and binomial distributions
- Estimation
- Hypothesis testing, $p$-values
- Linear regression
- Correlation coefficients
- Bayes theorem
However, I know my knowledge is far from complete. I ignore the mathematics of distributions such as the $\chi^2$ and $F$, I have zero knowledge on logarithmic regression, I have only a superficial idea of what ANOVA is...
Hopefully this provides a notion of where my knowledge and level of studies are at the moment. What should I read to go deeper?
It depends to some extent on what you mean by a "solid understanding". I will make some suggestions.
Less mathematical (mostly avoiding measure theory):
Statistical Inference by Casella and Berger. This is a standard late undergraduate to early graduate textbook.
A Course in Mathematical Statistics by George Roussas. Another text covering basic probability, point estimation, hypothesis testing, linear models, ANOVA, and so on. (He also has two probability and statistics books more basic than this, and a measure-theoretic probability textbook as well.)
More mathematical:
Mathmatical Statistics by Shao. Heavy use of measure theory. There is an accompanying Exercises and Solutions book.
Theoretical Statistics: Topics for a Core Course by Keener. This might be the most modern book here. It's the newest one, at least (published in 2010).
Theory of Statistics by Schervish. Possibly the most mathematically sophisticated book here. More Bayesian than the others, I think.