I know that many Statistics PhD students come from undergraduate / master's level math / applied math backgrounds. So, many will have taken a full-year of introductory real analysis and most likely also complex analysis (and lots of linear algebra and linear algebraic courses, such as scientific computing) -- but perhaps not measure-theoretic analysis.

Would you say that it is imperative to try and take a measure theory course to strengthen one's application to PhD programs in Statistics?

A lot of my classmates are fixated on the idea that measure theory is a necessity for Phd admission, while one professor that I spoke with internally thinks that this is a myth. But, he is a mathematician, so I'd rather ask the question here, just to gather more information.


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    In the U.S. at least, your first year course work is almost certain to include one year of graduate level probability theory. The class is likely to progress through discrete time measure and probability, to questions in continuous time. As a non stats student, I took this sequence at my school. Prior knowledge certainly helps, but is unlikely to be required. It's been a while, but we worked through Williams' "Probability with Martingales," then about 1/3-1/2 of Karatzas/Shreve. Maybe have a look if Williams strikes you as accessible. – gnometorule Apr 4 '16 at 10:32
  • Hi @gnometorule...so would the classes you took be the bare minimum of measure theory -- essentially knowing the lingo and the results needed to understand graduate probability? Or, was a pretty decent chunk of your course formal measure theory stuff? Thanks, – User001 Apr 4 '16 at 10:43
  • Also, I've read through some of Shreve's work, I forgot the title of his book. But he had a tiny chapter on the very basics of measure theory that would be needed to understand the rest of his book @gnometorule – User001 Apr 4 '16 at 10:46
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    We did measure theory from scratch. You go over it a bit fast to then use it in probability, but the first few weeks were mostly just basic discrete time measure theory. To follow along, all you needed was a good grasp of common analysis concepts (but fairly basic, it's all about sups and infs, really). No guarantee it's the same everywhere, but it was then how this was commonly taught. K/S (2nd quarter) has an entire long first chapter on continuous time measure theory, essentially establishing what a brownian motion is. Sounds very different from shreve's book you mention. – gnometorule Apr 4 '16 at 10:53
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    Ah, ok got it. Thanks so much for your helpful comments, @gnometorule. – User001 Apr 4 '16 at 11:15

Measure theory is essential for higher level probability.

However, I believe most graduate schools will have a graduate course on measure theory. Thus it is not absolutely necessary to have learnt measure theory before entering graduate school.

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    I would add also that the OP says statistics and not probability. While the fields are linked they are not necessarily the same. A statistics PhD could take a much more applied approach of examining data sets and coming up with new techniques to examine them that don't require a complete grounding in measure theory. Not that I would not recommend doing a course in it if it is an option! – Christy Apr 7 '17 at 9:33

Most statistics doctoral programs don't require their applicants to know measure theory, but then again, some very good doctoral statistics programs don't even require measure theory at all. (Case in point is Virginia Tech, which at least back in the early 2000's when I was there, had a great applied statistics department and measure theory was not required for the PhD program).

Do you need measure theory do statistics? Well, if you plan to work in academics as a statistician or want to do cutting edge research in financial analysis, you probably need it but I would say for a large portion of applied statisticians working outside of academics, the answer is no. IMO, it is not that important for the vast majority of what applied statisticians do (experimental design, data visualization, and estimation). Most of my friends who have high level positions in companies as statistical consultants report they never use it (which has been true for me also).

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