# rigorous statistical-inference book recommendation

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 that I would not recommend is "Mathematical Statistics" by Shao. In my opinion this book takes formality too far and attempts to subsume everything within measure theory, which I don't think is a particularly useful approach. "The Theory of Point Estimation" and "Testing Statistical Hypotheses" by Lehmann seem to be quite popular. – dsaxton Mar 31 '16 at 13:23
• There are several good recommendations here: stats.stackexchange.com/questions/33197/… For you, especially Young and Smith, Essentials of Statistical Inference. – kjetil b halvorsen Mar 31 '16 at 13:47
• Then how about first probability (Probability and Measure by Billingsley), then statistical inference (Trilogy by Lehman: Theory of Point Estimation, Testing Statistical Hypotheses, Elements of Large Sample Theory)? – Zhanxiong Dec 11 '17 at 4:53

## 3 Answers

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.