To first provide some context, I have two boxplots (displaying median, 25th, and 75th percentiles), and I'm wondering what a boxplot of the difference between these two boxplots would look like. The random variables (RV's) aren't necessarily distributed normally. Specifically, the "random variables" are samples from posterior distributions, but I think my question is ultimately pretty generic.
Since I have many samples from these 2 RV's, one way to do this would be to compute the differences between randomized pairings, then compute the quantile of interest from these differences. Because my statistical terminology might be flawed, let me say this in the R language:
X <- rnorm(10, 1000) # The first RV, a numeric vector of length 1000 Y <- rnorm(20, 1000) # The second RV Qs <- quantile(X-Y, probs=c(0.25, 0.5, 0.75) #The quantiles of their difference
However, I was wondering if there could be a way to compute the quantile of the difference between these RV's without going back to the original data. For example, from some combination of summary statistics (means, medians, variances, other quantiles, etc) computed from X and Y.
An example of something that does not give the correct answer, but which might shed light onto what I'm looking for:
wrong_Qs <- quantile(X, probs=c(0.25, 0.5, 0.75)) - quantile(Y, probs=c(0.25, 0.5, 0.75))
I am looking for something akin to the formulas for the propagation of uncertainty (variance) , but I haven't found anything yet.
The more I think about this, the more I'm thinking that there isn't a way to do what I'm asking. Any thoughts?