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)

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 on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



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.

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)