Covariance of natural logarithms — how to estimate $\sigma (\mathbf X, \mathbf Y)$ from $\sigma (\ln{\mathbf {(X)}},\ln{\mathbf {(Y)}})$?

Question

I have

$$\operatorname{Cov} (f{\mathbf {(X)}},f{\mathbf {(Y)}})$$

where $\operatorname{Cov}$ denotes the covariance and $f(\mathbf X)$ is a nonlinear function, i.e. $f(\mathbf X) = \ln(\mathbf X)$. Is there some transformation to estimate the covariance of the underlying variables? Such as:

$$\operatorname{Cov} (\mathbf X, \mathbf Y)$$

The background: I am estimating the expected error between two datasets. One has an error covariance matrix $\mathbf{S_1} = \operatorname{Cov}(\mathbf X,\mathbf Y)$, the other has an error covariance matrix $\mathbf{S_2} = \operatorname{Cov}(\ln \mathbf X, \ln \mathbf Y)$. To estimate the covariance for the comparison ensemble, I need to somehow add those two up; but how do I convert the covariance of the natural logarithms to the covariance of the quantity itself?

Answer 1

Using the general approach in this answer relating to covariance of transformed random variables may lead to a suitable answer.

More directly, using the results derived there, if $U=\log(X)$ and $V=\log(Y)$ ($\log$ being the natural log, naturally), the analysis in that question gets us to (after some rearrangement):

$$\operatorname{Cov}(X,Y) \approx \operatorname{E}(X)\operatorname{E}(Y)\left(e^{\operatorname{Cov}(U,V)}-1\right)\,.$$

This approximation is exact for $U,V$ bivariate normal. If the bivariate distribution on the log-scale is reasonably close to normal it should work quite well, especially in large samples.

However, further away from bivariate normality we need to keep in mind that there are several stages of approximation involved (indeed, the Taylor series expansion may not even converge), so some investigation may be required -- at least some simulation would help identify whether it might be useful on your problem.

As noted in that answer, other approximations can be arrived at by working the problem slightly differently; some of those may be more suitable for other distribution-cases.