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I know people love to close duplicates so I am not asking for a reference to start learning statistics (as here).

I have a doctorate in mathematics but never learned statistics. What is the shortest route to the equivalent knowledge to a top notch BS statistics degree and how do I measure when I have achieved that.

If a list of books would suffice (assuming I do the exercises lets say), that's terrific. Yes, I expect working out problems to be an implicit part of learning it but I want to fast track as much as realistically possible. I am not looking for an insanely rigorous treatment unless that is part of what statistical majors generally learn.

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    $\begingroup$ what field of mathematics you received your doctorate? This might be relevant. $\endgroup$ – mpiktas Jan 25 '11 at 19:42
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    $\begingroup$ Could you share with us why you want to learn stats? Curiosity? Needed for a project or research? Wanting to change jobs? Need to teach some courses? Want to collaborate with statisticians as the theoretical person? $\endgroup$ – whuber Jan 25 '11 at 19:51
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    $\begingroup$ I think it's almost always important to develop domain-specific expertise as well. A lot of statistics is learning the models relevant to specific fields. $\endgroup$ – Tristan Feb 22 '11 at 0:30
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    $\begingroup$ Try reversing "statistician wants equivalent knowledge to a quality maths degree" - there is not likely to be any fast routes. $\endgroup$ – probabilityislogic Aug 27 '13 at 10:12
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    $\begingroup$ "I know people love to close duplicates" made me laugh. $\endgroup$ – Mustafa S Eisa Dec 20 '16 at 20:06
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(Very) short story

Long story short, in some sense, statistics is like any other technical field: There is no fast track.

Long story

Bachelor's degree programs in statistics are relatively rare in the U.S. One reason I believe this is true is that it is quite hard to pack all that is necessary to learn statistics well into an undergraduate curriculum. This holds particularly true at universities that have significant general-education requirements.

Developing the necessary skills (mathematical, computational, and intuitive) takes a lot of effort and time. Statistics can begin to be understood at a fairly decent "operational" level once the student has mastered calculus and a decent amount of linear and matrix algebra. However, any applied statistician knows that it is quite easy to find oneself in territory that doesn't conform to a cookie-cutter or recipe-based approach to statistics. To really understand what is going on beneath the surface requires as a prerequisite mathematical and, in today's world, computational maturity that are only really attainable in the later years of undergraduate training. This is one reason that true statistical training mostly starts at the M.S. level in the U.S. (India, with their dedicated ISI is a little different story. A similar argument might be made for some Canadian-based education. I'm not familiar enough with European-based or Russian-based undergraduate statistics education to have an informed opinion.)

Nearly any (interesting) job would require an M.S. level education and the really interesting (in my opinion) jobs essentially require a doctorate-level education.

Seeing as you have a doctorate in mathematics, though we don't know in what area, here are my suggestions for something closer to an M.S.-level education. I include some parenthetical remarks to explain the choices.

  1. D. Huff, How to Lie with Statistics. (Very quick, easy read. Shows many of the conceptual ideas and pitfalls, in particular, in presenting statistics to the layman.)
  2. Mood, Graybill, and Boes, Introduction to the Theory of Statistics, 3rd ed., 1974. (M.S.-level intro to theoretical statistics. You'll learn about sampling distributions, point estimation and hypothesis testing in a classical, frequentist framework. My opinion is that this is generally better, and a bit more advanced, than modern counterparts such as Casella & Berger or Rice.)
  3. Seber & Lee, Linear Regression Analysis, 2nd ed. (Lays out the theory behind point estimation and hypothesis testing for linear models, which is probably the most important topic to understand in applied statistics. Since you probably have a good linear algebra background, you should immediately be able to understand what is going on geometrically, which provides a lot of intuition. Also has good information related to assessment issues in model selection, departures from assumptions, prediction, and robust versions of linear models.)
  4. Hastie, Tibshirani, and Friedman, Elements of Statistical Learning, 2nd ed., 2009. (This book has a much more applied feeling than the last and broadly covers lots of modern machine-learning topics. The major contribution here is in providing statistical interpretations of many machine-learning ideas, which pays off particularly in quantifying uncertainty in such models. This is something that tends to go un(der)addressed in typical machine-learning books. Legally available for free here.)
  5. A. Agresti, Categorical Data Analysis, 2nd ed. (Good presentation of how to deal with discrete data in a statistical framework. Good theory and good practical examples. Perhaps on the traditional side in some respects.)
  6. Boyd & Vandenberghe, Convex Optimization. (Many of the most popular modern statistical estimation and hypothesis-testing problems can be formulated as convex optimization problems. This also goes for numerous machine-learning techniques, e.g., SVMs. Having a broader understanding and the ability to recognize such problems as convex programs is quite valuable, I think. Legally available for free here.)
  7. Efron & Tibshirani, An Introduction to the Bootstrap. (You ought to at least be familiar with the bootstrap and related techniques. For a textbook, it's a quick and easy read.)
  8. J. Liu, Monte Carlo Strategies in Scientific Computing or P. Glasserman, Monte Carlo Methods in Financial Engineering. (The latter sounds very directed to a particular application area, but I think it'll give a good overview and practical examples of all the most important techniques. Financial engineering applications have driven a fair amount of Monte Carlo research over the last decade or so.)
  9. E. Tufte, The Visual Display of Quantitative Information. (Good visualization and presentation of data is [highly] underrated, even by statisticians.)
  10. J. Tukey, Exploratory Data Analysis. (Standard. Oldie, but goodie. Some might say outdated, but still worth having a look at.)

Complements

Here are some other books, mostly of a little more advanced, theoretical and/or auxiliary nature, that are helpful.

  1. F. A. Graybill, Theory and Application of the Linear Model. (Old fashioned, terrible typesetting, but covers all the same ground of Seber & Lee, and more. I say old-fashioned because more modern treatments would probably tend to use the SVD to unify and simplify a lot of the techniques and proofs.)
  2. F. A. Graybill, Matrices with Applications in Statistics. (Companion text to the above. A wealth of good matrix algebra results useful to statistics here. Great desk reference.)
  3. Devroye, Gyorfi, and Lugosi, A Probabilistic Theory of Pattern Recognition. (Rigorous and theoretical text on quantifying performance in classification problems.)
  4. Brockwell & Davis, Time Series: Theory and Methods. (Classical time-series analysis. Theoretical treatment. For more applied ones, Box, Jenkins & Reinsel or Ruey Tsay's texts are decent.)
  5. Motwani and Raghavan, Randomized Algorithms. (Probabilistic methods and analysis for computational algorithms.)
  6. D. Williams, Probability and Martingales and/or R. Durrett, Probability: Theory and Examples. (In case you've seen measure theory, say, at the level of D. L. Cohn, but maybe not probability theory. Both are good for getting quickly up to speed if you already know measure theory.)
  7. F. Harrell, Regression Modeling Strategies. (Not as good as Elements of Statistical Learning [ESL], but has a different, and interesting, take on things. Covers more "traditional" applied statistics topics than does ESL and so worth knowing about, for sure.)

More Advanced (Doctorate-Level) Texts

  1. Lehmann and Casella, Theory of Point Estimation. (PhD-level treatment of point estimation. Part of the challenge of this book is reading it and figuring out what is a typo and what is not. When you see yourself recognizing them quickly, you'll know you understand. There's plenty of practice of this type in there, especially if you dive into the problems.)

  2. Lehmann and Romano, Testing Statistical Hypotheses. (PhD-level treatment of hypothesis testing. Not as many typos as TPE above.)

  3. A. van der Vaart, Asymptotic Statistics. (A beautiful book on the asymptotic theory of statistics with good hints on application areas. Not an applied book though. My only quibble is that some rather bizarre notation is used and details are at times brushed under the rug.)

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    $\begingroup$ @cardinal, ex-Soviet universities have separate undergraduate statistics studies. In Vilnius University for example you can get bachelor degree in statistics. From what I see with the students I wholeheartedly agree that master or even doctorate-level education is needed for interesting jobs. $\endgroup$ – mpiktas Feb 22 '11 at 7:13
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    $\begingroup$ @cardinal, @mpiktas 4 years in BS + 2 yearts in MS + 4 years in PhD makes ten years to learn something interesting :) I would give $+\infty$ to this great answer if possible. Most books are new to me. $\endgroup$ – Dmitrij Celov Feb 22 '11 at 10:41
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    $\begingroup$ @John Salvatier, you're correct that those methods aren't covered in this text. Then again, this strikes me as more a matter of taste, particularly since the text's main focus is not on algorithms. To wit, your concerns are directly addressed by the authors in the introduction (pg. 13). $\endgroup$ – cardinal Feb 22 '11 at 17:49
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    $\begingroup$ @cardinal: Scandinavian universities usually do offer bachelors level degrees as well. That being said, I think statisticians take themselves a bit too seriously. I disagree that you'd need a doctorate degree in order to get an "interesting" job. I believe that as science and research becomes more and more cross-disciplinary statistics have been imposed on studies from many different areas. Half of the articles on high impact journals have some questionable statistical analysis, just to meet the demands, even though it might not make any sense, given in original context/domain of the problem. $\endgroup$ – posdef Feb 24 '11 at 12:52
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    $\begingroup$ @cardinal Mood book was a great suggestion because nowadays it is difficult to find an introductory book on statistics that is formal enough for someone with a math background. Has anybody read this book new book? Panateros, "Statistics for Mathematicians" springer.com/us/book/9783319283395 $\endgroup$ – Igor Fobia Nov 12 '18 at 14:30
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I can't speak for the more rigorous schools, but I am doing a B.S. in General Statistics (the most rigorous at my school) at University of California, Davis, and there is a fairly heavy amount of reliance on rigor and derivation. A doctorate in math will be helpful, insomuch as you will have a very strong background in real analysis and linear algebra--useful skills in statistics. My statistics program has about 50% of the coursework going to support the fundamentals (linear algebra, real analysis, calculus, probability, estimation), and the other 50% goes towards specialized topics that rely on the fundamentals (nonparametrics, computation, ANOVA/Regression, time series, Bayesian analysis).
Once you get the fundamentals, jumping to the specifics is usually not too difficult. Most of the individuals in my classes struggle with the proofs and real analysis, and easily grasp the statistical concepts, so coming from a math background will most definitely help. That being said, the following two texts have pretty good coverage of many topics covered in statistics. Both were recommended in the link you provided, by the way, so I wouldn't say your question and the one you linked are necessarily uncorrelated.

Mathematical Methods of Statistics, by Harald Cramer

All of Statistics: A Concise Course in Statistical Inference, by Larry Wasserman

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    $\begingroup$ +1 All of Statistics: it would be a great place to start. $\endgroup$ – Simon Byrne Feb 21 '11 at 23:20
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    $\begingroup$ the UC-Davis program looks good and I think you'll get a great education there. I wouldn't consider it "less rigorous" than other places. I thought the comment on their "integrated B.S./M.S. degree" page was interesting and relevant to the thread: "There is a high demand for statisticians, but the knowledge and skill achieved by those with a BS degree in Statistics are often not sufficient for the needs in the [government or industrial] workplace." $\endgroup$ – cardinal Feb 26 '11 at 23:57
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The Royal Statistical Society in the UK offers the Graduate Diploma in Statistics, which is at the level of a good Bachelor's degree. A syllabus, reading list, & past papers are available from their website. I've known mathematicians use it to get up to speed in Statistics. Taking the exams (officially, or in the comfort of your own study) could be a useful way to measure when you're there.

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    $\begingroup$ The Graduate Diploma exams are effectively final year undergraduate exams; for "staging" purposes there are lower level certificates that can be taken first. The RSS exams are available, if I recall correctly, worldwide with the exception of Hong Kong (which has its own statistical society and exams). An alternative is the undergraduate Diploma in Statistics offered by distance learning by the Open University in the UK, but again available worldwide. This is of slightly lower level than the RSS Grad Dip so may be seen as preparation for it. As a taught course it's substantially more expensive. $\endgroup$ – Silverfish Nov 5 '13 at 3:28
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I would go to the curriculum websites of the top stats schools, write down the books they use in their undergrad courses, see which ones are highly rated on Amazon, and order them at your public/university library.

Some schools to consider:

Supplement the texts with the various lecture video sites such as MIT OCW and videolectures.net.

Caltech doesn't have an undergrad degree in statistics, but you won't go wrong by following the curriculum of their undergrad stats courses.

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    $\begingroup$ this seems like a bit of an odd list. To my knowledge, Carnegie Mellon is the only school on that list that (formally) offers an undergraduate degree in statistics. Neither Caltech nor MIT even have graduate programs in statistics. $\endgroup$ – cardinal Feb 22 '11 at 0:07
  • $\begingroup$ @cardinal. why must you doubt me? :) I put in links to the undergrad stats courses at those fine institutions. Also, mixing and matching courses from the best schools will beat out following a degree path from a worse school. $\endgroup$ – Neil McGuigan Feb 22 '11 at 0:28
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    $\begingroup$ OCW is certainly a very fine resource and a great initiative. This is no doubting that. As for your assertion that mixing and matching from the "best schools" is a superior solution, I find that highly suspect, particularly for undergraduate studies. While a highly motivated student is bound to get a very good undergraduate education at any of those schools, an undergrad education as good or better can be found at many, many "worse" schools. Schools such as those that you list do tend to "win out" for graduate education, I would say. $\endgroup$ – cardinal Feb 22 '11 at 1:40
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    $\begingroup$ Actually, this was the first thing I tried. I tried this before posting the question. Finding a list of courses wasn't hard, but finding information about which books actually got used for those courses and what sections of those books were covered was much more difficult. $\endgroup$ – John Robertson Feb 24 '11 at 21:02
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I have seen Statistical Inference, by Silvey, used by mathematicians who needed some workaday grasp of statistics. It's a small book, and should by rights be cheap. Looking at http://www.amazon.com/Statistical-Inference-Monographs-Statistics-Probability/dp/0412138204/ref=sr_1_1?ie=UTF8&s=books&qid=1298750064&sr=1-1, it seems to be cheap second hand.

It's old and concentrates on classical statistics. While it's not highly abstract, it is intended for a reasonably mathematical audience - many of the exercises are from the Cambridge (UK) Diploma in Mathematical Statistics, which is basically an MSc.

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Regarding the measurement of your knowledge: You could attend some data mining / data analysis competitions, such as 1, 2, 3, 4, and see how you score compared to others.

There are a lot of pointers to textbooks on mathematical statistics in the answers. I would like to add as relevant topics:

  • the empirical social research component, which comprise sampling theory, socio-demographic and regional standards
  • data management, which includes knowlegde on databases (writing SQL queries, common database schemes)
  • communication, how to present results in a way the audience stays awake (visualization methods)

Disclaimer: I am not a statistician, this are just my 2cents

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E.T. Jaynes "Probability Theory: The Logic of Science: Principles and Elementary Applications Vol 1", Cambridge University Press, 2003 is pretty much a must-read for the Bayesian side of statistics, at about the right level. I'm looking forward to recommendations for the frequentist side of things (I have loads of monographs, but very few good general texts).

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    $\begingroup$ I would suggest It's a must read for anybody who wants to be a good statistician, Frequentist, Bayesian or anything else. $\endgroup$ – probabilityislogic Jan 26 '11 at 0:16
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    $\begingroup$ I disagree, Jaynes' book is a terrible recommendation in this circumstance: 1) the notation is sloppy and non-standard, which makes it difficult to cross reference with other sources, 2) he's long winded and gets bogged down in silly and irrelevant arguments (the OP asked for the "shortest route") 3) there's also the errors (such as the marginalization paradox) $\endgroup$ – Simon Byrne Feb 21 '11 at 22:56
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    $\begingroup$ @Dikran Marsupial, do you own the Schervish text on statistical inference? I've been on the fence regarding whether to purchase it or not, so was curious, since you appear to align yourself pretty strongly with the Bayesian approach. $\endgroup$ – cardinal Feb 22 '11 at 19:07
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    $\begingroup$ I wouldn't say I was strongly aligned to the Bayesian approach. It is the approach I understand the best, which is not the same thing. Essentially I am an engineer at heart, and I want both tools in my toolbox, maintained in good order! A proper understanding of the benefits and disadvantages of each approach is what we should aim for. I haven't got Shervishes book, but I did read a paper of his on Bayes factors that seemed quite flawed to me (I'll see if I can find it and post a question for someone to explain it to me!). $\endgroup$ – Dikran Marsupial Feb 22 '11 at 19:24
  • $\begingroup$ @Dikran, Your (potential) question sounds interesting. I look forward to a post on it. $\endgroup$ – cardinal Feb 24 '11 at 0:56
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I come from a computer science background focusing on machine learning. However, I really started to understand (and more important to apply) statistics after taking a Pattern Recognition course using Bishop's Book https://www.microsoft.com/en-us/research/people/cmbishop/#!prml-book

here are some course slides from MIT:
http://www.ai.mit.edu/courses/6.867-f03/lectures.html

This will just give you the background (+ some matlab code) to use statistics for real work problems and is definitely more on the applied side.

Yet, it highly depends on what you want to do with your knowledge. To get a measure for how good you are you might want to browse the open course ware of some university for advanced statistics courses, to check if you know the topics covered. Just my 5 cent.

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I think Stanford provides the best resources when it comes to flexibility. They even have a machine learning course online that would provide you with a respectable base of knowledge when it comes to designing algorithms in R. Search it up on Google and it will redirect you to their Lagunita page where they have some interesting courses, most of them being free. I have Tibshirani's books, Introduction to Statistical Learning' and 'Elements of Statistical Learning' in PDF formats and both are extremely good resources.

Since you're a mathematician, I would still advise you to not fast track as that wouldn't provide you with a solid base that you might find very helpful in the future if at all you start doing some serious machine learning. Treat statistics as a branch of mathematics for getting insights from data, and that requires some work. Other than that, there are tons of online resources, Johns Hopkins provides similar stuff as Stanford. Although experience always pays, a respectable credential will always reinforce that base. You can also think of the specific fields that you would like to enter; by that I mean whether you want to go into text analytics or applying your math and stats skills in finance. I come in the latter category so I have a degree in econometrics where we studied finance + statistics. A combination can always be very good.

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