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I'm fairly mathematically inclined — had 6 semesters of Math in my undergrad — though I'm a bit out of practice and slow with say partial differential equations and path integrals my concepts come back with a bit of practice. I have not had a course on mathematical proofs (mathematical thinking) or one on analysis.

I also understand graduate level probability — have studied it formally and refreshed my knowledge lately.

I also have had a couple of graduate level courses on statistics and statistical learning.

I want to, out of personal interest, study mathematical statistics over the next 18-24 months. I'd like to devote an average of 5 hours a week of self study to the subject.

I am a bit at a loss on how to do it. I tried studying from the Casella and Berger book but really could not make any headway. I found the book a bit boring and its method intractable.

What I found difficult about Casella and Berger:

  1. Embarrassed to say this but the starting for the type setting — the way it was packed to reduce white space wore me down
  2. There are a lot of proofs that were there but I felt there was a lack of intuition on why we were trying to achieve the results and what was the larger goal at hand.
  3. The referencing of proofs from previous chapters was in a way that made the material a bit intractable to me — I was going back a lot until I finally gave up.
  4. The example seemed to be very doable however I could not tackle the problems — the problems seemed to be in a class by themselves.
  5. I just could not get into the material — and I wonder if the way my mind works I need a more rigorous treatment — should I consider a measure theoretic approach to mathematical statistics?

So question: is there a textbook that someone in my shoes could just study off of and teach themselves the subject.

What I would like in a text:

  1. In many ways the stuff I'd like in a book are the inverse of things I didn't like in Casella and Berger.
  2. The type setting of the book would help. Some of the points below will elaborate this point.
  3. I think it would be good to have a book which starts of with an intuition on what we would like to do, perhaps in a non-mathematical sense — somewhat like the book Statistics by Freeman et al.
  4. A book that presents the theorems in a simultaneous mathematical derivation and commentary format — In CB, I just gave up on trying to read up on the proofs
  5. A book that has a good range of solved problems accompanying each section.
  6. A book that also has computational exercises that allow the reader the build a better understanding by exploring the concepts say using R
  7. A book that covers the material that would be required for the first one or possibly two graduate courses in Mathematical Statistics.

Additional Notes:

  1. I am aware of this question Introduction to statistics for mathematicians — and there is some overlap and some of the answers I've studied before posting this question — however I feel that the two questions have different asks.
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    $\begingroup$ It's not at all clear the way in which Casella and Berger doesn't suit you, which means alternatives anyone might suggest may be even worse. There's very little basis on which to guess what might be 'ideal' for you. $\endgroup$ – Glen_b Jul 7 '15 at 9:46
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    $\begingroup$ Your edit is a definite improvement because it gives some pointers about what you don't want. Your post went through a review process after your edit and several of our users voted to leave it closed, suggesting that the feeling is that the question is still too broad. If possible I'd suggest an additional edit to clearly identify things you seek in a book, but if anyone wants to weigh in on what information they'd like to see, here please do so. $\endgroup$ – Glen_b Jul 8 '15 at 0:03
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    $\begingroup$ Many thanks @Glen_b -- I'll give it a shot -- I have been thinking of what would make a text more suitable for self study $\endgroup$ – user975917 Jul 8 '15 at 0:06
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    $\begingroup$ If you have not taken any undergraduate Analysis, might this be a problem if you want to go down the measure theory path? Depending on your background, it sounds like that would require some additional preparation. $\endgroup$ – Silverfish Jul 8 '15 at 1:45
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    $\begingroup$ @Silverfish, I agree -- I'm not sure how much preparation a measure theoretic course requires. $\endgroup$ – user975917 Jul 8 '15 at 2:20
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On the grounds that you want something (a) well-motivated, (b) less dense, and (c) introductory (undergraduate or early graduate level), you might want to consider a text like "Mathematical statistics and its applications" by Larsen and Marx. The "and its applications" is important because the authors give a practical motivation to the theory that you may have found missing in Casella and Berger. This is still a "mathematical statistics" book though, not an applied practitioner's guide on how to apply statistical methods that are otherwise treated as a "black box". There are exercises in Minitab, which I am sure you could translate into another statistical language of your choice.

It only covers a small fraction of what C&B do, and it may not be "pure" enough for your tastes; perhaps you will find the applications a sort of contamination rather than motivation! But C&B is quite a heavy book to hit, if it's the first that you take on. Larsen and Marx is (in my opinion) quite clearly written, covers simpler material, and is very well type-set. That all should make it easier to get through. Perhaps after working through a book pitched at this level, it would be easier to mount a second assault on C&B or similar.

The reviews on amazon are pretty mixed; it's interesting that people who taught courses using the book were generally pretty favorable (one criticism is that it is not as mathematically rigorous as it might have been) while students on courses where the book was a set text were more negative.

If you would prefer a text that was more mathematical in nature, then I think you might need to work on your background knowledge first. I can't see how it is possible to understand a rigorous proof of the Central Limit Theorem without a good background in analysis, for instance. There are some "intermediate" texts, of which Larsen and Marx is one, which are not so rigorous as to be incomprehensible to someone without an analysis background (so you get a "sketch proof" of the CLT rather than a formal one, for example), but which are still "mathematical statistics" rather than "applied statistics". I suspect your basic choice lies between the more mathematical approach, or reaching into statistics via this sort of intermediate-level book. But if you want to take things higher, then at some point you are going to need some more mathematics.

MIT runs a course for introductory statistics for (undergraduate) economics, with a set text of "Probability and Statistics for Engineers and Scientists" by Sheldon Ross, and recommended texts of Larsen and Marx or alternatively DeGroot and Schervish, "Probability and Statistics". The MIT course authors compare them as:

Larsen and Marx's book is a bit more chatty than Ross', while DeGroot and Schervish's is a very good book but somewhat more difficult

If you want something antithetical to the dry style of C&B then the chattier style of L&M might suit you. But those other suggestions for texts of a similar difficulty level might also interest you.

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  • $\begingroup$ Many thanks for your notes @Silverfish, I'll give L&M a good look -- I think I looked at it but was turned off by the reviews. But from how you formulate the differences between the books L&M may be it for me. I've gone over a lot of the Sheldon Book -- from what I remember it was an introductory probability books -- unless we're talking about two different books. $\endgroup$ – user975917 Jul 8 '15 at 6:54
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    $\begingroup$ The very first course I took on statistics many years ago, was based on (earlier ed of) Larsen & Marx. Given that the poster has some stats courses, that would seem to elementary a book! $\endgroup$ – kjetil b halvorsen Aug 8 '17 at 23:01
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    $\begingroup$ @kjetil I did wonder if it might be too introductory. I think someone else commented here that they found the book useful though that comment since seems to have vanished (perhaps I am confused with another thread). But what L&M seems to me to do very well is to combine proof with motivation. And in some cases that's just an outline proof, but without e.g. studying Analysis first, I think that's unavoidable (another reason I thought the book might be a good fir the OP). $\endgroup$ – Silverfish Aug 8 '17 at 23:10
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    $\begingroup$ @kjetil What also influenced my thinking is that it's quite easy to take a lot of stats courses, even up to postgraduate levels, that are really "applied data analysis" courses with no mathematical theory/justification whatsoever. The OP didn't clarify what their courses covered (though if their program had been pushing the mathematical side, I'd have expected Analysis to have been a compulsory prerequisite), but they did seem to be looking for something introductory. Someone who has that kind of background will still find L&M a step up mathematically, but an easier one than C&B. $\endgroup$ – Silverfish Aug 8 '17 at 23:17
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    $\begingroup$ Sheldon here perhaps means [Sheldon] Ross (or indicates overdosing on Big Bang Theory). $\endgroup$ – Nick Cox Feb 1 '18 at 1:34
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For me, Hogg & Craig has always worked as my second reference and back-up for those moments when Casella & Berger didn't make much sense to me. While both are excellent and share more or less the same scope, I found the former easier to read (it has more textual explanations on how the formulae work) and the latter a bit more dry with the mathematics (maybe too economical with the derivations).

I totally suggest you give this book a try and see if it fits your needs!

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I agree that it might be easier to answer this question with a little bit more about what you're looking for. However, after CB I would recommend Grimmett and Stirzaker and Wasserman's All of Statistics. G&S has a nice accompaniment with worked problems, so plenty of excitement there.

Best of luck!

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    $\begingroup$ Many thanks for your response -- I'm considering the G&S book -- I've added detail to my question -- perhaps it will allay some of your concerns. $\endgroup$ – user975917 Jul 7 '15 at 16:37
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    $\begingroup$ I wouldn't recommend Grimmett and Stirzaker because it's probability rather than statistics (as far as I remember). $\endgroup$ – mark999 Jul 8 '15 at 1:46
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    $\begingroup$ I posted very early before the asker added a great deal of clarity to his/her question. All of Statistics provides an excellent compact collection of results/proofs. For select fundamental topics like set theory, RVs, and convergence G&S is solid with the added bonus of a companion book with worked solutions. Although yes there is no inference, it could be handy. $\endgroup$ – nooreen Jul 8 '15 at 3:43
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The following are both a step down from Casella-Berger in terms of the level of detail they go into, but are rigorous enough that they are used as introductory graduate textbooks. They're both well presented and fairly recent. Plus they're different enough from each other in layout and content that you could read them in parallel without too much duplication:

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  • $\begingroup$ Strongly agree on Rice. An excellent grounding in the main ideas is accompanied by a strong awareness that statistics is also about data analysis. $\endgroup$ – Nick Cox Jan 31 '18 at 18:32
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Given that the OP has had some course in statistics and probability, maybe something like https://www.amazon.com/Mathematical-Statistics-Basic-Selected-Topics/dp/0132306379 the second edition of Bickel & Doksum's book (there is also a volume 2!). This book is maybe not very rigorous, but it includes many very modern ideas, especially from nonparametric statistics.

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