Distribution of the Spearman rank correlation coefficient under the assumption of non-zero correlation There are some papers and some R packages providing exact calculations for the CDF and the inverse-CDF of the (sample) Spearman rank correlation coefficient.
My question is: How difficult would it be to calculate the exact distribution of the Spearman rank coefficient of correlation for a given setting like this (just an example):

*

*$X$ and $Y$ continuous with known marginal distributions (not necessarily normal).

*$X$ and $Y$ are correlated with a known (population) Pearson correlation coefficient different from zero.

*Of course, the value of the sample size $n$ is known, and, since both $X$ and $Y$ are considered to be continuous, the non-ties case can be assumed.

If not, I was thinking of estimating the distribution by using Monte-Carlo simulation. What do you think? Any other idea?
By the way, I know that the Spearman coefficient is a distribution-free method, meaning that its probability distribution does not depend on the concrete distribution of $X$ and $Y$ under the (null) hypothesis of independence. But I think it's clear that its distribution does depend on the  concrete distribution of $X$ and $Y$ under the (alternative) hypothesis of non-zero correlation (as it happens with many distribution-free methods, anyway).

It's not possible to do this exactly, as knowing the marginal distributions and a correlation coefficient is not sufficient to determine the joint distribution, which would be necessary to do this. Even knowing it, however, would probably not help you in practice, since Spearman is a rank correlation, so you would have to convert the joint distribution of $X$ and $Y$ to a joint distribution of the ranks of a sample of size $n$, which seems to me to not be a practical thing to do for any significant $n$ and almost any distributions, especially continuous ones.
However, given the joint distribution, simulation becomes (in many practical cases) a feasible alternative, although requiring that you abandon the goal of knowing the exact distribution under the alternative.  Of course, if you have a point null and a point alternative hypothesis, the Spearman correlation coefficient is not likely to be the best statistic to use to discriminate between them, especially given the Neyman-Pearson lemma.
