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I am asking for a reference in basic statistics following, in the same spirit as my similar question for probability here. That is it should:

  • Assume some mathematical maturity and can make use of analysis, linear algebra, etc from page 1. On the other hand it should assume nothing itself on statistics itself.
  • Be concise but still show examples. It should present the basics results of statistics (whatever that means, I am trying to say that I prefer a book focusing on few fundamental things).
  • Not sloppy, but I am not looking for particularly abstract or research oriented viewpoint.

What is it for: I would like to get some basic understanding of machine learning and other data science related techniques. Before doing so, I would like to get some basics in statistics. The rough amount of time I would like to dedicate to this is roughly 6-8 weeks. I am not pretending to learn everything which can be useful for data science in such a short time, but rather the most essential things which would allow me to have a look ahead with some real understanding, rather than just following recipes and copy-pasting code found somewhere.

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  • $\begingroup$ Check this one. [link]people.math.umass.edu/~lavine/Book/book.html $\endgroup$ – Vikranth Inti May 21 '17 at 9:55
  • $\begingroup$ In spite of your best efforts to explain your situation, your question is somewhat vague. But even if it were more precise, such a question is likely to draw a vast variety of quirky individual answers. My own personal quirky and possibly useless suggestion is Bain and Englehardt: Intro to Prob and Math Stat, perhaps still available for a reasonable price in paperback. (My only defense for making the suggestion is that I have taught out of this book several times and found it to cover a broad range of topics in a mathematically rigorous fashion.) $\endgroup$ – BruceET May 22 '17 at 15:19
  • $\begingroup$ Well, unfortunately I do not think I can improve the question, it is one of those cases in which if I knew exactly what I need I would not need to ask. Still, I am hoping that a list of quirky suggestions on math.stackexchange can be more useful than a list of Google generated results... $\endgroup$ – GFR May 22 '17 at 18:44
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    $\begingroup$ "look ahead with some real understanding, rather than just following recipes" Like freshman calculus, statistics is a subject that is introduced as a large number of "recipes" more or less. Once you've seen a good scattering of methods for statistical estimation and hypothesis testing, the attraction of a unifying theory ("statistical probability") will either wax or wane for you. Without the supporting evidence of many and varied examples, the theory of statistics is remarkably arcane (as math subjects go). $\endgroup$ – hardmath May 22 '17 at 19:22
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Weighing the Odds by David Williams is a good introduction to statistics by someone who mainly works in probability. That makes it much more interesting to read because he emphasizes some of the mathematical aspects that are interesting. The beginning of the book is a concise introduction to probability theory (without measure theory), with just enough to be able to study statistics.

Since your other question asked about probability theory itself, if Williams' book doesn't serve your purpose, you could consider Probability: An Introduction by Grimmett and Welsh, which is also fairly concise.

Obviously, there are also books that use measure theory, but they're much more involved.

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I'd also like to find a concise presentation of statistics for mathematicians who are already comfortable with linear algebra and analysis. I don't know a book that exactly fits this description, but here are a few books I've found that seem useful:

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To get a good understanding of data science, how to apply the methods one needs a comprehensive practical view of statistics. You are on the right track when you say you don't simply want to copy paste code. Here is a collection of books that will guide you through that learning process.

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First let me note that I found the comment by hardmath very appropriate: even given the ability to follow a mathematically sophisticated presentation, it is probably worth getting some acquaintance to the field and be exposed to examples first.

I had to settle on some references relatively quickly so I could not explore those suggested much, but hopefully I will found the time for this at some point. For now I am mostly using the book "Statistical Inference" by Casella and Berger which treats both probability theory and statistics, and is neither too sloppy nor too advanced.

I plan to read some parts of either "The elements of statistical learning" by Hastie et al, or the simpler, but with example in the programming language R, "An introduction to statistical learning" by James et al.

The programming languages most wort learning first in data science seem to be R or Python, so I will probably integrate with a book or online course exploring application using either. Machine learning by Andrew NG (Coursera I think) is very famous. Unfortunately it uses some other programming language but maybe is still worth following.

I may have some more time in the future to peruse other suggested references, and other people may be interested, so do not let this answer stop you from posting yours!

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