I have three random variables (T, O, A) that are each approximately normal. These three random variables are combined to form two ratios: X=T/A and Y=O/A. I wish to get an estimate of the covariance between X and Y (i.e. between the two ratios (T/A) and (O/A). I suspect that this could be done using a Taylor expansion but I can't work out how.
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1$\begingroup$ ssc.wisc.edu/cde/cdewp/72-18.pdf Pearson (1897) answered this question $\endgroup$– user112516Commented Apr 17, 2016 at 16:27
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Try using the delta method with $f(x,y,z)=(x/z,y/z)$. You need to calculate the derivative of this mapping and substitute for the general formula. This will give you the asymptotic covariance matrix of the two ratios. See for example https://en.wikipedia.org/wiki/Delta_method#Multivariate_delta_method.