confidence interval of a difference of variables Let's say I got two samples from two unknown random variables: 
$(x_i), (y_i)$
At this stage I don't know (or don't want to assume) that they come from a similar process or not.
I want to compute a confidence interval of the expected difference $E[Delta] = E[X - Y]$.
Given the large size of the samples I suspect we can assume that the expected difference would be gaussian.
Therefore I imagine I can use the standard approach that builds the CI like this: $\mu +/- z^\alpha\sigma$ where $z^\alpha$ is the value of the inverse gaussian CDF at the given confidence level.
Where I'm pretty stuck is basically how to estimate the mean and standard deviation of the expected difference $E[X - Y]$.
 A: Normally when estimating a confidence interval I think of an interval estimate for a population parameter (such as the mean). So when you say you want to compute a confidence interval for the difference, I assume you mean an interval estimate for the expected difference.  Is this correct?  
whether or not X-Y is Gaussian depends on the distribution of X and Y.  If both X and Y are Gaussian, than X-Y will be Gaussian.  if not, than X-Y will, in all likelihood, not be Gaussian. 
When you're bootstrapping I am guessing that on each iteration you store the sample average difference.  So you're really developing an interval for the estimate of $E(X-Y)$.  Lets call this estimate $\bar{d}$.  
In the parametric approach $\bar{d}$  will be the sample average of $X-Y$.  $\bar{d}$ WILL be normally distributed by the central limit theorem, regardless of the distribution of $X$ and $Y$ assuming they are both iid.  So even though X-Y is not Gaussian, $\bar{d}$ will be (at least in the limit as the number of observations approaches infinity).
Assuming $X$ and $Y$ are iid, $X-Y$ will have mean $E(X-Y)=E(X) - E(Y)$ which can be approximated by the sample average $\bar{d}$.  The variance of this estimate can be derived in the following matter.
$$
var(\bar{d}) = var(\frac{1}{n}\sum_{i=1}^{n}(x_i - y_i))=...=\frac{\sigma^2}{n}
$$
Where $\sigma^2=var(X-Y)=var(X)+var(Y)-2cov(X,Y)$.  So roughly speaking
$$
\bar{d} \sim n(E(X-Y) , \frac{\sigma^2}{n})
$$ 
and more formally
$$
\sqrt(n)(\bar{d}-E(X-Y)) \rightarrow n(0,\frac{\sigma^2}{n})\;\;\;(in\;distribution)
$$
Of course we don't know $\sigma^2$ so we instead approximate it with the sample variance $s^2$.
You can then form a confidence interval for $E(X-Y)$ with the above Gaussian distribution and it will hold asymptotically. I suppose you could also use the t-distribution in developing the confidence interval, I would not know off hand what the degrees of freedom would be (best guess n-1).
Hope that helps! 
