an inverse permutation test

September 22, 2016
By

[This article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

A straightforward but probabilistic riddle this week in the Riddler, which is to find the expected order of integer i when the sequence {1,2,…,n} is partitioned at random into two sets, A and B, each of which is then sorted before both sets are merged. For instance, if {1,2,3,4} is divided in A={1,4} and B={2,3}, the order of 2 in {1,4,2,3} is 3. An R rendering of the experiment is

m=rbinom(1,n,.5)
if (m*(n-m)>0){
fist=sort(sample(1:n,m))
return(order(c(fist,sort((1:n)[-fist])))[i])
}else{
return(i)}
[\sourcecode]

It is rather easy to find that the probability that the order of i takes the value j is

{i-1 \choose j-1}(1/2)^i

if j

{n-i \choose n-j}(1/2)^{n-i+1}

if $j>i$ (i is in B), the case i=j being the addition of both cases, but the mean can be found (almost) immediately by considering that, when i is in A, its average value is (i+1)/2 and when it is in B, its average value is (n+i)/2 [by symmetry]. Hence a global mean of (2i+n+1)/4….

Filed under: Books, Kids, R, Statistics Tagged: order statistics, permutation tests, R, sorting, The Riddler

To leave a comment for the author, please follow the link and comment on their blog: R – Xi'an's Og.

R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Search R-bloggers

Sponsors

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)