Real Analysis and Statistics What level of real analysis do you think is desirable for the study of statistics?
I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am not far off. I have always considered analysis a little annoying because of its many self-evident results. I will study it nevertheless if it can make you a better statistician. What is your opinion? Thank you.
 A: I have studied both analysis and statistics to some extent.
The short answer to your question is that a good understanding of an undergraduate book on real analysis (titles such as 'An Introduction To Real Analysis') should be enough. Attempting the exercises, understanding the answers and being able to talk about nearly everything in the (chosen) book with ease is a good sign of having sufficient knowledge of real analysis.
However that is not the full story. 
In my view, an understanding of real analysis to a much deeper level is not needed to be able to do 'good'  or 'simple' statistical work. But it will immensely help for many reasons.
In statistics there is never one problem. Complexity is always there. Often there are many smaller problems in one problem. This decreases intuition because there is no one sole task to be completed. Often solutions to the smaller problems fight with each other - the Type I error $\alpha$ and Type II error $\beta$ in hypothesis testing, for example.
Not many mathematicians use the word 'rigorous'. It is non-mathematicians who (usually) do so. To a mathematician, a proof is a proof. This concept of rigour is an excuse for introducing intuition to shorter an explanation. 
Intuition is far more important in statistics than mathematics. Real analysis is responsible for formalising intuition into 'rigour'. Everyone knows that as the number $x$ increases, the number $1/x$ decreases. That's intuition. Real analysis gives you rigour.
Consider the following example. 
Let us measure the response times of people in some situation where the response time, $s$, is between $0$ and $1$ seconds. Suppose that you want to know the exact response time - you want a measuring tool to be able to find the response time and you use this as your criterion
The label $y$, will take value $1$ if the response $s$ is an rational number (such as, say $0.25$ or $1$) and will take value $0$ if the if the response $s$ is an irrational number (such as, say $1/\pi$).
To a statistician this is trivial - the experiment is not so exciting. But a mathematician well trained in real analysis will immediately see the problems here - in fact it is a disaster of an experiment.
You will want to be able to analyse the results of this experiment on a computer, say, and publish your findings. But you will not be able to:


*

*Publish your findings

*Do any efficient computer implementation

*Find a measuring tool to do so

*Understand 1., 2., 3. in the first place or be able to solve the problems.


Real analysis is the answer. Specifically, the set of non-computable numbers forms a strict subset of the transcendental numbers, in say, the continuum $[0,1]$. Then almost all (as the set of rationals, in the continuum $[0,1]$ has measure zero) responses should be irrational. As a result, almost all responses are transcendental and almost all responses are non-computable. This means they cannot be represented by a well-defined algorithm on a computer. This means that cannot be measured.
Also, the example, when attempted to be plotted, resembles the Dirichlet function. It cannot be plotted - there are no responses on the horizontal values $0$ and $1$. It is due to the density (real analysis) that the responses can take.
It seems like a silly example, yes, but it is not explained by statistics. It is explained by real analysis and more generally, mathematics. The real headache would be not understanding the problem in a 'rigorous' manner.
