What is the distribution of the difference of two iid noncentral Student t variates Let $X_1$ and $X_2$ be iid non-central t random variables.
I'm interested in the question: 
what is the distribution of $X_1 - X_2$?
i.e. what is the distribution of the difference of two iid noncentral Student t variates?

Suppose $d$ is an observed estimate for either $X1$ or $X2$, in R code, the likelihood function for $d$ will be:
likelihood = function(x) dt(d*sqrt(N), df, ncp = x*sqrt(N)) 
where d = an observed estimate of X1 or X2, x = parameter range (-Inf to Inf), N = sample size, and df = N - 1.
P.S. dt(x,df,ncp) is the pdf of a noncentral t distribution with the third argument ncp being the non-centrality parameter.
 A: Looks like I am a little late.  Anyway, as per Owen (D.B. Owen, “A Survey of Properties and Applications of the Noncentral t distribution”, Technometrics 10 (1968) 445-478), if x is noncentral t distributed and $\nu > 2$, then
$$var[x] = \frac{\nu}{\nu-2}+\delta^2[\frac{\nu}{\nu-2}-\frac{\nu}{2}\frac{\Gamma^2((\nu-1)/2)}{\Gamma^2(\nu/2)}]$$
where $\nu = df$ and $\delta = NCT$ is the noncentrality parameter.
Using $\nu = 10$ and $\delta = 5$, var[x] = 3.1386 so the variance of the difference of two of these is 6.2773.  I generated $10^7$ of these differences and binned them into a histogram, shown below.  The variance of the $10^7$ differences was 6.2779.  Unfortunately, I have no idea what function the histogram approximates.

A: Since you use R and don't need an exact solution, you may find the distr package for R useful, at least for exploring.
For fixed degrees of freedom and non-centrality parameter you can start exploring with code like:
library(distr)

d1 <- Td(df=10, ncp=5)
d2 <- Td(df=10, ncp=5)

plot(d1)

dd <- d1 - d2
plot(dd)

I am not sure how to incorporate the non-centrality depending on x.
