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I am self-studying statistics (I am a undergrad mathematician).

I learned all of my statistics from a mathematical statistics book based on measure theory. I learned (by "learned", I mean "read a book which contained one detailed chapter on this topic"):

  • Estimation theory (MLE + other methods + asymptotic results)
  • Hypotheses
  • Confidence intervals
  • Linear model (regression + classification)
  • Mixed models

Where should I go next? I prefer books, and as I am an advanced math student, I am fine with high mathematical requirements. I prefer them, probably. Or maybe I should now take a more applied approach?

I also know programming in R and python, so that's ok as well.

Jonah
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  • Two papers by people thinking broadly and deeply in the areas you express interest in are: Lin and Tegmark's *Critical Behavior from Deep Dynamics* (here ... https://ai2-s2-pdfs.s3.amazonaws.com/5ba0/3a03d844f10d7b4861d3b116818afe2b75f2.pdf) as well as J.P. Bouchaud's *Crises and Collective Socio-Economic Phenomena* (here ... https://www.cfm.fr/assets/ResearchPapers/Crises+and+collective+socio-economic+phenomena.pdf). Follow the bread crumbs in the references. – Mike Hunter Jan 16 '17 at 20:14
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    I would recommend time series analysis and bootstrap methods. For the bootstrap my book with Robert La Budde "An Introduction to Bootstrap Methods with Applications to R" is probably on your level and it has applications illustrated in R. This was published in 2011 by Wiley. For time series I like the books by Richard Davis and Peter Brockwell which are at your level and give a good treatment in both the frequency and time domains. – Michael R. Chernick Jan 16 '17 at 20:26
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    With all respect--many of us have extensive mathematical backgrounds--you haven't even begun to learn statistics. Your education is akin to trying to learn quantum mechanics by studying $C^{*}$ algebras. The analogy isn't a bad one: the mathematical study might prepare you to appreciate mathematical descriptions of physical experiments, but the next step would be to perform a set of classical lab experiments. Likewise, you shouldn't feel ashamed to begin exploring and analyzing simple datasets--and perhaps to study introductory stats textbooks to learn how. – whuber Jan 16 '17 at 21:11
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    I agree with both the analogy whuber makes, and the suggestion of looking at actually making contact with data (which is where the fun is). Theorems are important, there's no doubt of that, but they don't always help you figure out how and when to do what, what else you might do instead and how much practical difference the choice might make. They don't usually tell you what parts of a real problem are fine to abstract away and what parts really matter. Ultimately choosing what to learn next depends on what you want to be able to do, but that's hard without exploring the territory a bit first. – Glen_b Jan 16 '17 at 23:06
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    People learn at varying rates and via varying pathways. Not all learning has to be linear, slowly incremental and cumulative. It can happen in leaps and spurts as well as in the absence of a grand, overall plan. Who's to say what the best approach is for the OP? – Mike Hunter Jan 17 '17 at 14:57
  • A close dup: https://stats.stackexchange.com/questions/414/introduction-to-statistics-for-mathematicians – kjetil b halvorsen Sep 02 '18 at 11:00
  • Statistics is not necessarily applied statistics. Many statisticians who publish in Annals, JASA etc. barely ever do applied work. So if you prefer theory over applications that is absolutely fine. Moreover, although @whuber might think otherwise, mathematical statistics is not the same as mathematics but can be an interesting field of study on its own. Your biggest problem might be that by learnt you mean read, so I suppose you already know what you should do next: exercises! – user1587692 Mar 30 '19 at 13:50

2 Answers2

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I think the best way forward at this point is to decide where you want to specialize, start reading academic articles from the top journals, and familiarizing yourself with the methods you encounter. If you want to go medical, read the biostatistics journals like biometrika. If you want to go finance/econ, read econometrika.

Other topics in statistics

  • Generalized linear models
  • Stochastic processes and time series
  • Bayesian probability and estimation
  • Semiparametric estimation theory (Cox models, sandwich estimators, quasilikelihood)
  • Nonparametric estimation theory (minimax, smoothing)
  • Survival analysis
  • Survey design
  • Power analysis, simulation, and so on

Other great books McCullogh & Nelder: Generalized Linear Models. Boos and Stefanski: Essential Statistical Inference. Wakefield: Frequentist and Bayesian Regression Models. Agresti: Categorical Data Analysis. Hosmer Lemeshow: Logistic Regression Models, Hosmer Lemeshow & May Survival Analysis. Diggle Heagerty Liang & Zeger: Longitudinal Data Analysis. Sheldon Ross: Probability Models. Ferguson: A Course in Large Sample Theory.

AdamO
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Partially answered in comments:

With all respect--many of us have extensive mathematical backgrounds--you haven't even begun to learn statistics. Your education is akin to trying to learn quantum mechanics by studying C∗ algebras. The analogy isn't a bad one: the mathematical study might prepare you to appreciate mathematical descriptions of physical experiments, but the next step would be to perform a set of classical lab experiments. Likewise, you shouldn't feel ashamed to begin exploring and analyzing simple datasets--and perhaps to study introductory stats textbooks to learn how. – whuber

I agree with both the analogy whuber makes, and the suggestion of looking at actually making contact with data (which is where the fun is). Theorems are important, there's no doubt of that, but they don't always help you figure out how and when to do what, what else you might do instead and how much practical difference the choice might make. They don't usually tell you what parts of a real problem are fine to abstract away and what parts really matter. Ultimately choosing what to learn next depends on what you want to be able to do, but that's hard without exploring the territory a bit first. – Glen_b

( Other more specific advice left in the comments )

kjetil b halvorsen
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