Reasoning about an arbitrary transformation function based on observed data

I have two lots of data that have been independently, arbitrarily transformed from the same original data source using the same permutation algorithm. I would like to know what kind of statements I can make about the original data source or the transformation process by comparing these two data sets. All the data are non-negative integers.

To express this in R code:

# These true values and permutation function are unknown/unobservable
set.seed(1904)
true.xs <- round(rnorm(1000, mean=100, sd=15), 0)
unknown.trans <- function(orig)orig + round(rnorm(length(orig), sd=3),0)

# These observations have been made
xs.1 <- unknown.trans(true.xs)
xs.2 <- unknown.trans(true.xs)


I would like to know what I can say about true.xs or unknown.trans given xs.1 and xs.2. I'm willing to make some fairly strong assumptions if that will help (i.e. that the transformation function is symetrical around zero, etc.).

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Are you talking about permutations here, or about some other transformation of data? If you mean permutations, then your R code seems to be incorrect, because sum(true.xs == 96) is 29, and sum(xs.1 == 96) is 23. – Leo Sep 6 '12 at 18:22
Apologies, just some arbitrary transformation of the data. The permutation is used incorrectly in the original problem domain, but I'll change that now. – fmark Sep 6 '12 at 23:49