# Why would a statistician need to know how to do delta - epsilon proofs?

#### claudius

##### New Member
Many statistics PhD programs recommend that prospective students take a course in real analysis (undergraduate or graduate). While I understand that real analysis helps build mathematical maturity, is it even applicable in statistics (other than maybe measure theoretic probability)?

#### spunky

##### Can't make spagetti
hell yeah! i dont think anyone who goes into any mathematics-like area of knowledge (math, statistics, computer science, physics, etc...) should skip real analysis (or even complex analysis for that matter). it'll give you a direct insight into the foundations of statistics, help you learn how to evaluate a mathematical argument and... i dunno, things in higher-level courses just make a lot more sense after you've taken real analysis... **especially** at the PhD level where you're gonna have to end up proving a lot of stuff so you better know your basics if you're willing to venture into going there...

#### fed1

##### TS Contributor
I actually took math stats class before real analysis.

I think that you can learn a lot of stats theory without real analysis. Many theorems from real analysis etc, cauchy schwarz and friends for example, have specific representations in the math stats course that can be learned there "on the fly"".
Of course, you can take alot of analysis and not learn the most basic statistics theory like sufficiency etc..

On the other hand, many technical details about the mathematics involved in a Math Stats Course are only taken up in real analysis class.

#### Dason

But if you're going into a PhD program that values theory at least somewhat you will probably find yourself taking a course in measure theory to really dig down into the nitty gritty of the theory behind it all. If that's the case then you would be hopelessly lost without having real analysis beforehand.

#### Dragan

##### Super Moderator
I'd suggest that you take a look at Alan Polansky's book "Introduction to Statistical Limit Theory" and then I think you will find how asymptotic theory plays a critical role in statistics e.g. Convergence of Moments, Convergence of Random Variables, Asymptotic Expansions for Distributions and Random Variables, Central Limit theorems, etc...... In short, an understanding of Real Analysis is very important.

I think too many grad students get caught up in trying to spend all their time learning sophisticated statistical techniques. This diminishes their focus on pure mathematics.

#### spunky

##### Can't make spagetti
my god......maybe it's because i'm dealing with this as we speak but why *does* a stats student have to focus on pure mathematics..ever?
because you simply cannot call yourself a statistician (whether applied or theoretical) if you're unwilling or unable to follow through the rigour of formal logic and proof. you can be an informed quantitative methodologist, a skilled data analyst, or a good practitioner, etc. but not a statistician. not going through those courses would mean you can only end up giving hand-waving arguments an quasi-intuitive explanations as for why things work the way they work. this is not entirely bad in itself, but it is contingent on what you want to do in your professional life. when i took my first course in real analysis that's kind of what my prof said: "you're here to earn a degree in mathematics so i will treat you and expect you to behave like mathematicians" and bam! set theory...

i think half of the problem is the attitude. yeah, they're hard, but so what? just give it your best shot and study extra hard. now, if you cant follow the courses regardless how much you try... i dunno... why not consider a program that focuses on data analysis more and less on the theoretical aspects of math? there're always quantitative-oriented programs in social sciences depts at the unis if you look for them...

#### spunky

##### Can't make spagetti
statistics is a broad but still specifc instance of mathematics. there can be mathematics without statistics but there cannot be statistics without mathematics, so you're within the same stream most of us were trained here. so i dunno... maybe because i already did it, it does make sense to me. do keep in mind that if your program is housed in a statistics department you are, for all practical and theoretical purposes, a mathematician of sorts and you are expected to be trained as such. even if you love data analysis or the applied aspect of it, if you're going for a degree on anything statistsics then you have to be trained as a mathematician. which is why other quantitative-oriented programs have either the words "measurement" or "quantitative" or "applied" or some other thingy in front of them.

one thing that helped me out was taking lots and lots of notes, even if i didnt understand everything that was being said at first so that later, in the quiteness of the night, i would take them out and review them until it all started to make sense... or ask a bazillion questions! you're paying for this thing!

#### Dason

i chose the program because i love the material (outside of calc/mathstats), and i can't think of very many topics at all i would have loved to study for this long.
Stick with it! If you love it don't let a small sequence of courses ruin it for you. I hated my design of surveys course but at the end of it I was glad I took it. Not because of the material. Oh no - that class was worthless to me. But because I showed to myself that I could stick with something that was absolutely painful. A lot of people have that experience with real analysis. I love real analysis and my painful experience was with design of surveys.

With that said... the single most important statistics course I took while in my undergrad was Math Stat I&II (ok so it was two courses but you get the point). To me those courses connect pretty much everything you've learned previously and put it on a solid ground. This is good because it actually gives you the ability to figure out if what you're doing is appropriate or not. It also allows you provides you with the tools to deal with situations you may not have encountered before. This might not be you want to do (figuring out new methods) but it's still very useful.

I'm a stats grad student so I'm slightly biased. I love the theory I'll admit it. But being able to think mathematically is an important skill. It might be painful to get through but keep your head up and I'm sure you'll get through it.

#### BGM

##### TS Contributor
And from my experience is that the stuff you learn right now may not look useful for your career at all, but if you want to work in the academic field, there is a high chance that you will meet those stuff again sooner or later. Not only to math Stat (of course it is important and I admit I love this course too), but all subjects related to stat.

#### spunky

##### Can't make spagetti
is calculus all that's bothering you? i mean, it's good if it's only one that thing because then you know where to go for the attack... i'm not sure about the content of the course but assuming it was similar to mine, can you work through the linear algebra? is it easy for you to follow the logic behind how the derivations or proofs are constructed?

#### Dason

i don't know what to say.... its just so remote from everything.
Hmm. This sounds like part of the problem! Math Stats isn't remote from everything. It's intertwined with everything in statistics! I don't know if it's the book you're using or if your professor is having a hard time getting that across and making connections to the applications but they're there. Now it might be that your course is very mathematical and in that case they might not bother to try to connect the theory to anything but it's there. You can always ask questions here if you want some motivation if you have specific questions.

And just so you know - it's probably my guess that not everybody here had everything come easy. It takes a long time to learn and sure it might come easier to some but it still took a lot of work. I know I personally struggled with quite a few things.

#### spunky

##### Can't make spagetti
I know I personally struggled with quite a few things.
that's a lie and we know it...

ok, nah, on a more serious tone, i think we all had our moments where we just didnt wanna go on. but hey, such is life, and school is not the only place where it happens. you get up and try again and again and again and put more and more and more effort until eventually you crack it...

ps- but dason's right. mathematical statistics is where you start seeing the connections... i'm thinking, you mentioned that you need to take a stream of 2 courses in math-stats. what's course #2 on? maybe there's something there that intersests you and motivates you to just be done with course #1

#### Dason

Also I know we've talked about how you don't like asking questions here but... we're here to help if you want/need it! Whether it's asking for some motivation for a problem or just getting a push in the right direction we'll be happy to help.

#### Dason

Once again if you need help with the calculus you can post here! There are a few tricks to a lot of the integrals you'll deal with that aren't quite so obvious at first but once you get the hang of it everything becomes a lot easier. We can point out a few of those tricks

Also - what book are you using?

#### CB

##### Super Moderator
I'm enjoying this thread. In psychology in NZ we get a little training in data analysis but really nothing about the mathematical theory underlying statistics. So while I know a bit about complicated-sounding things like structural equation modelling at least in terms of clicking buttons and writing nice stories about the output that results, I'm lost with even the simplest of the more mathematical problems that crop on the TS boards. For a long while I've been wondering about how much value there would be in trying to take some courses in basic mathematical statistics. I'm not really meant to take extra courses on top of the full-time PhD but there must be some way... My career aim isn't to be a statistician, I'm more aiming to be a researcher or academic in the social sciences specialising in quantitative methodology, but I'm still wondering if it might be worth doing. Comments?

##### Ninja say what!?!
If you want to be in academics, I would recommend taking the courses.

If it's mainly in research, I'd say it depends. Would the understanding help you in doing better research?

#### spunky

##### Can't make spagetti
For a long while I've been wondering about how much value there would be in trying to take some courses in basic mathematical statistics
take them.. take them all! do a 2nd degree if you need to! we need more people who can understand what happens with this magic box of SPSS, R or Mplus, where all we know is " data goes in, p-values comes out ". besides, you'll get a few cracks in the meantime. the first time i encountered factor analysis i couldnt help but laugh a little bit when they showed me what these "factors", the "latent variables that driiiiiiive the correlations are" .... i.e. they're nothing more than the eigenvectors of the covariance matrix. for reasons beyond my understanding such eigenvectors have been worshipped as gods in this province of knowledge. but once you know what they are and where they come from, you can actually work with them in very interesting ways.

#### CB

##### Super Moderator
take them.. take them all! do a 2nd degree if you need to!
Heh. After switching from Finance, to clinical psych, then out of clinical to a general psych PhD, I think a complete new degree in a different subject would be stretching it a bit far But I think I might start hunting for some papers I could take!