most recent 30 from http://mathoverflow.net 2013-05-22T19:20:52Z http://mathoverflow.net/feeds/question/31655 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31655/statistics-for-mathematicians Nokker 2010-07-13T03:26:06Z 2010-12-26T15:03:08Z

<p>I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of statistics.</p> <p>Most texts I've come across are either too basic, or are monographs focused on a specific area or technique.</p> <p>Any suggestions?</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31665#31665 robin girard 2010-07-13T05:56:01Z 2010-07-13T05:56:01Z

<p>The Book of Van der Vaart is exaxcly what you are searching for ! Asymptotic statistics <a href="http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521784504" rel="nofollow">http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521784504</a>. </p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31671#31671 T. 2010-07-13T07:52:43Z 2010-07-13T07:52:43Z

<p>Wasserman's <em>All Of Statistics</em> seems concise and useful. </p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31673#31673 Andrew L 2010-07-13T08:04:20Z 2010-07-13T08:04:20Z

<p>It's old fashioned,but it's written by a master and most of it may just be what you're looking for: <em>Mathematical Statistics</em> by S.S.Wilks. It's long out of print and hard to find,but worth hunting down. </p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31675#31675 Mikael Vejdemo-Johansson 2010-07-13T08:45:52Z 2010-07-13T08:45:52Z

<p>I'm currently working my way through <em>Cramér</em>'s <em>Mathematical methods of statistics</em>. It starts out with a half-book primer on all measure theory, Lebesgue integration et.c. you might possibly need for anything, and then goes through first probability and then statistics with this backdrop.</p> <p>Skipping the introductory analysis it might be what you're looking for. </p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31687#31687 PeterR 2010-07-13T11:47:12Z 2010-07-13T11:47:12Z

<p>I liked Mathematical Statistics by Jun Shao for the classical theory. There is also a companion 'Problems &amp; Solutions' book. </p> <p>I second Wasserman's 'All of Statistics' for a quick overview.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31707#31707 madhadron 2010-07-13T14:16:01Z 2010-07-13T14:16:01Z

<p>Kiefer's "Introduction to Statistical Inference" is a particularly nice book. It crams an incredibly amount of perspective into a very easy to read package. You'll want to supplement it with volumes like Lehmann's "Point Estimation" and "Testing Statistical Hypotheses."</p> <p>One thing I particularly like is that he starts at the bottom with the decision theory structure, and imposing other criteria to be able to choose from the plethora of admissible procedures before diving into things like linear estimators and hypothesis tests.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/31755#31755 Michael Hardy 2010-07-13T19:40:18Z 2010-07-13T22:37:41Z

<p>This won't directly answer the question, but here are some things a mathematician who wants to learn about statistics should learn:</p> <ul> <li>When is a random variable a statistic and when is it not? (A statistic is an <em>observable</em> random variable. For example $X - E(X)$ is not a statistic if the "population average" $E(X)$ is not observable.</li> <li>Fisher's concept of sufficiency. Examples, characterizations, theorems. In particular, the Rao--Blackwell theorem and examples of its use. That's way cool.</li> <li>So is the concept of completeness and the Lehmann--Scheffe theorem.</li> <li>If you think that linear regression is called linear because you're fitting a line, then you are naive. If you're fitting, e.g., a parabola by finding least-squares estimators of three parameters, then you're doing linear regression. There is also such a thing as non-linear regression.</li> <li>Learn the Gauss--Markov theorem on Best Linear Unbiased Estimators (BLUEs).</li> <li>Look at my recent answer to a question on prediction intervals. Why do you need the (finite-dimensional version of) the spectral theorem to understand linear regression? (Look at the aforementioned answer and consider this question an exercise.)</li> <li>As long as we're on linear regression (the topic of the three bullets immediately above this one), look at the Wikipedia article titled "errors and residuals in statistics" (written mostly by me). Learn the difference between an error and a residual. Maybe look at "Studentized residual" as an afterthought.</li> <li>....and then at "lack-of-fit sum of squares".</li> <li>If you think linear regression is child's play rather than something to which the most brilliant person could devote a long career in research, grow up.</li> <li>Learn the difference between frequentism and Bayesianism. In fact, look at the rant I posted on nLab about this. (The essence of Bayesianism is that probabilities are taken to be epistemic. Bayesianism is not more subjective than frequentism; rather Bayesians and frequentists put their subjectivity in different places. (A really glaring example is the 5% critical value legendarily used in medical journals. Why 5%? Because that's a subjective economic choice.))</li> <li>Learn design of experiments. Learn why Latin squares and a myriad of other combinatorial designs are used.</li> <li>OK, maybe a small and incomplete but nonetheless direct answer to the original question: perhaps Hocking's book on linear models.</li> <li>Learn to use the word "sample" correctly. If you ask the next 100 people you meet whether they intend to vote "yes" or "no", that's not 100 samples; that's one sample.</li> <li>Another thing that will give you some idea of the distinct flavor of the subject, and how it differs from probability theory and some other fields, is books on sampling.</li> <li>Learn about the Wishart distribution.</li> <li>And the multivariate normal distribution.</li> <li>Exercise: How do you prove that every non-negative-definite matrix is the variance of some random vector?</li> <li>Learn why the Behrens--Fisher problem cannot be regarded as a math problem. It belongs up there with Hilbert's problems as one of the great challenges, but it's not mathematics for this reason: One can model it as a math problem in any of a variety of different non-equivalent ways. One can solve those math problems. But which one is the "right" model? That's essentially a philosophical question. And that question, not the math problems, is the Behrens--Fisher problem. (The Behrens--Fisher problem is this: how do you draw inferences about the difference between the means of two normally distributed populations which may have different variances? "Inferences" can mean point-estimates or interval estimates or perhaps other things.)</li> </ul> <p>This is just a sampling of the first things that come immediately to mind. It leans toward showing you what the subject tastes like rather than what it's important to know to do theoretical or applied research.</p> <p>Statistics is an immensely broader field than mathematical probablity theory.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/49043#49043 Pascal Orosco 2010-12-11T13:14:45Z 2010-12-11T13:14:45Z

<p><em>Probability Theory: the Logic of Science</em> by physicist E. T. Jaynes is a must.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/49044#49044 Joël 2010-12-11T13:32:38Z 2010-12-11T13:32:38Z

<p>There is a book I love: "Statistical inference" by S.D. Sylvey. Short, elemntary, but perfectly clear and suited for the mathematical mind. Impossible not to like statistics after that.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/49047#49047 Gil Kalai 2010-12-11T14:22:44Z 2010-12-11T14:22:44Z

<p>I like "Statistics" by Freedman, Pisani, Purves, and Adhikari</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/50338#50338 David Epstein 2010-12-25T14:18:56Z 2010-12-25T14:18:56Z

<p>Joël : the book you love is by S.D. Silvey, not Sylvey.</p> <p>David Williams "Weighing the Odds" is a book about probability and statistics by a distinguished probability theorist. He has a great sense of humour and the book is a lot of fun. The exercises can be demanding, but they are also interesting. (I have latex solutions to many of them if you want to ask me after first trying an exercise.) Professional statisticians think this book is more maths than stats, but it does contain a lot of stats. It's an introduction that will hold your attention if you put effort into the exercises. It's aimed at advanced undergraduates, but in my opinion is also very suitable for research mathematicians (though I have my doubts as to its suitability for a professional probability theorist).</p> <p>After reading and enjoying this book, a mathematician will find it much easier to get to grips with a good book on stats. But there's no substitute for trying to handle large amounts of real data, when nothing works out as it "ought" to. Learning stats without handling data is like learning maths without working out anything for yourself---it's too easy to kid yourself that you understand.</p>

http://mathoverflow.net/questions/31655/statistics-for-mathematicians/50427#50427 aginensky 2010-12-26T15:03:08Z 2010-12-26T15:03:08Z

<p>For a very mathematical version of statistics, my favorite is on line lecture notes from two MIT courses. The instructor is named Panchenko and the course is called 'Statistics for Applications'. There are course notes that read like a book for the course in 2003 and 2006. I have enjoyed browsing through both of them. Here is a link: <a href="http://ocw.mit.edu/courses/mathematics/" rel="nofollow">http://ocw.mit.edu/courses/mathematics/</a>. </p>