If we know that random variables $X$ and $Y$ are positively correlated, do we know that the random variables $Z = \log(X)$ and $W = \log(Y)$ are positively correlated?
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2 Answers
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1
No, without any additional assumption, knowing the (Pearson) correlation on $X$ and $Y$ does not give any clue on the (Pearson) correlation between $\log X$ and $\log Y$. See the following example in R:
x1 = c(10^-100, 1, 10^5)
x2 = c(1, 10^-100, 10^5)
cor(x1, x2) # = 1
cor(log(x1), log(x2)) # -0.4251781
(Here, $X_1$ and $X_2$ can take $3$ values with equal probabilities $\frac{1}{3}$.)
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1$\begingroup$ +1 for the nice counter example. But this would be clearer if you started with "No" rather than "Yes". $\endgroup$ Commented Jan 23, 2013 at 5:54
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We don't.
You can calculate an approximation via Taylor Series that should work fairly well for X and Y having a small coefficient of variation or being close to normal.
