Formula for confidence intervals for corelation of samples from non-normal distributions

My question is similar to Formula for 95% confidence interval for $R^2$ (I indeed want to corelate predicted with true values to obtain $R^2$).

The problem is their distribution is not normal. I wanted to use Cohen et al. (2003) approach. However I dig through Correlations redux Olkin and Finn (1995) and The mean and second moment coefficient, in samples from a normal population Wishart (1931), from which the formula originates. Wishart states explicitely:

[...] althought the utility of this quantity, for a distribution which is far from normal, is not so great as would at first sight appear.

Is there known a formula for such CI, or do I have to resort to bootstrap-like approach?

• The formula, as you already note, would have to depend on the distribution: there is no universal formula, just as there isn't even a universal formula for the CI of a mean. Would you like to provide some specific distributional assumptions?
– whuber
Feb 7 '18 at 21:56
• Unfortunately the only thing I can assume is that number of samples is large (>> 100). In case of mean I think it may be enough due to the central limit theorem. But I have no idea how to use it in case of corelation. Feb 7 '18 at 22:06
• @whuber I may also assume that both variance and mean of distributions are finite Feb 7 '18 at 22:25
• That's not going to get you very far. In fact, you already made that assumption by supposing $R^2$ exists.
– whuber
Feb 7 '18 at 22:44
• @whuber then it seems bootstrap is the only option Feb 8 '18 at 13:26