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I am currently preparing for a qualifying exam in mathematical statistics and am looking for good resources for self-study. I have access to old exams which I have been slowly working on; however, I find that I do not know a lot of the "tricks" to solve problems more efficiently and usually take the long way to get to a solution. Most of the tricks I do know came from the courses I took, but that is only a limited set that I do not feel will be enough for my qualification exam.

Currently I am using Statistical Inference by Casella & Berger as my main reference.

I looked at other posts for reference material but none seem to be evaluated based on example quality. Working out problems and reading theorems is fine, but it would be helpful to have problems with solutions or a text with rigorous, well-explained examples.

What are some of your recommendations?

User1865345
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    it would be helpful to have problems with solutions or a text with rigorous, well-explained examples In many regards, this is why the theorems have proofs given, not to convince you that the claim is correct (that’s necessary in the primary literature, probably less so in a textbook) but to show how to work through showing a result. Thinking this way might help you better utilize your Casella/Berger book: the theorems could have been given as exercises, and the proofs in the book serve as a solution manual for them. // I’ve found a Casella/Berger solution manual. It seems to exist somewhere. – Dave Aug 31 '23 at 04:05
  • @Dave, I agree with you and C&B is an excellent book for an undergrad as the solutions are already provided by the authors. Unfortunately, they lack the measure theoretic language as well as many important topics (either briefly mentioned/under-developed or not mentioned at all) that a graduate student should know. – User1865345 Aug 31 '23 at 04:10
  • IIRC Casella Berger has a solutions manual. – AdamO Aug 31 '23 at 04:56
  • @User1865345 A 1st year theory exam would not require measure theory, though folks coming from a strong math background might be ready for Fergusson (A Course in Large Sample Theory) or Lehmann Casella (Theory of Point of Estimation). – AdamO Aug 31 '23 at 05:19
  • @AdamO, oh, I see. In any case, Ferguson is also a good material considering he provided solutions. Speaking of that, I would also suggest looking at his book on decision theory which is also solved by the author. – User1865345 Aug 31 '23 at 05:44
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    The bottom page of this study guide in probability & statistics. – stans Aug 31 '23 at 06:40

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$\bullet$ Have a look at Examples and Problems in Mathematical Statistics by Shelemyahu Zacks. It is a revamp of a previous book by the same author Parametric Statistical Inference Basic Theory and Modern Approaches.

What I like about the book is it invests proper time in rigorously elaborating every minor aspect of the concept in concern and lays out the measure theoretic language in an accessible and lucid manner. Mainly it contains some really good examples and ample exercises most of which the author has has provided solutions or sufficient hints.

$\bullet$ Another book I would say would be apt after reading C&B is Theoretical Statistics: Topics for a Core Course by Robert W. Keener. It is less measure theoretic but it is a moderate level book that covers most of the topics in decision theory, classical and Bayesian statistics that a typical graduate student should be acquainted with. While it has no vast collection of examples or problems, the latter are pretty much decent and the author does provide solutions to most of them.

I prefer a rigorous, formal, measure-theoretic development but in an accessible tone without being too concise (in that note, if you want, you can check Mathematical Statistics by Jun Shao, but I am not too much enthusiastic or inclined on using this book as a first course of study pertinent to the style I mentioned; nevertheless, there are decent collections of problems, all of which are solved by the author in a separate book) - the two books above cater to my purpose. The problems aren't exhaustive and in no way can one claim that solving them would make you proficient. But those would certainly help, for sure.

Before delving into them, please note that you do have taken a paper on measure theory.

$\bullet$ Another book that I forgot to mention (courtesy AdamO) is Thomas S. Ferguson's Mathematical Statistics: A Decision Theoretic Approach. This is a decision-theoretic book built heavily on the previous books on decision theory by Wald; Blackwell, Girshick. It is an intermediate level book with little to no measure theory. However, Ferguson is known for his plain straightforward writing style and is replete with standard examples. Selected exercises are solved by the author and he still maintains it in his homepage.

The one that AdamO mentions is another book by the same author on large sample theory, which also contains decent examples and problems, but personally I felt the book is bit condensed from the theory point of view.

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