Thursday, December 12, 2019

A Math-Stat Conjecture: Please Help Prove?

Are you strong in math stats or probability?  I have a conjecture.  It is about order statistics.  The result is a bit surprising and I haven't figured out why it is so beautiful.

First,  an easy problem.  Suppose
\[X_i \stackrel{iid}{\sim }\mbox{G},\; i=1,\ldots,I\]
is a sequence of independent and identically distributed continuous random variables with some distribution \(G\).  For example, \(G\) could be a standard normal distribution.  And suppose the same for \(Y_i\):
\[Y_i \stackrel{iid}{\sim }\mbox{G},\; i=1,\ldots,I.\]
Further, define \(A=\{i: \mbox{ such that } X_i<Y_i\}\).  Let \(n\) be the size of \(A\), that is, n is the the number of times \(X_i<Y_i\).  It should come as no surprise that \(n\) is distribution as binomial with a probability of .5 and a size of \(I\), and this holds regardless of \(G\).  So, for example, if \(I=5\), then the probability distribution of \(n\) is:




That is the easy problem.  Here is the hard problem.  Suppose we do the same, but instead  if comparing the samples, we sort  them first so  \(X_1<X_2<\dots<X_I\) and \(Y_1<Y_2<\ldots<Y_I\).  Then, my contention is that \(n\) is distributed as a discrete uniform with mass \(1/(I+1)\) on points \(0,\ldots,I\).  The following is some R code:

I=5
M=100000
n=1:M
for (m in 1:M){
  X=sort(rnorm(I))
  Y=sort(rnorm(I))
  n[m]=sum(X<Y)}
p=table(n)/M

Here is the graph of the probability mass function:



Cool result.  I just can't prove it.  Any help?  Be a math-stat hero.  Thanks!


6 comments:

Matthias Ossadnik said...

Thanks for posting this interesting problem!

If you are still interested, I posted a proof here

https://mossadnik.github.io/2020/01/10/sorted-random-variables.html

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