Suppose I have $X,Y$ which are lognormal with correlation, $\rho$, what is the correlation between $log(X)$ and $log(Y)$? I tried working it out analytically and I'm getting that you have to approximate is using a taylor series. Is that correct?
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1Assuming you intend that $(\log X, \log Y)$ be bivariate normal, then the answer here should get you most of the way: https://stats.stackexchange.com/questions/132855/expectation-variance-and-correlation-of-a-bivariate-lognormal-distribution/132879#132879 ... you just need to write that covariance term on the right hand side in the equation at the end as a correlation times the two lognormal standard deviations, and simplify (there's some cancellation). Beware the difference between the notation there (where $\rho$ is the conventional log-scale correlation) and what you have above. – Glen_b Aug 19 '19 at 23:10
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1There's also a related answer here: https://stats.stackexchange.com/questions/6853/correlation-of-log-normal-random-variables though it may be a little harder to see how to back it out from that one. Nevertheless it uses the same relationships as the first link. – Glen_b Aug 19 '19 at 23:11
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@Glen_b Thanks, basically, I just want to simulate this in python. Do you know if I can compute the correlation and then use this: https://stackoverflow.com/questions/31977175/how-can-i-sample-a-multivariate-log-normal-distribution-in-python. I want this method in python (https://www.rdocumentation.org/packages/MethylCapSig/versions/1.0.1/topics/mvlognormal). – John Doe Aug 19 '19 at 23:37
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The answer to the question you link to describes the usual approach (generate MVN and then exponentiate). – Glen_b Aug 19 '19 at 23:43
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@Glen_b so would I need the formula you linked to get the correlation between the normals and then use that approach? – John Doe Aug 19 '19 at 23:46
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You would need it for the function you linked to as well, since the argument to that function is the log-scale correlation. – Glen_b Aug 19 '19 at 23:48