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).