Consider the CV for a given random variable $X$:
$$ CV_X = \frac{\sigma_X}{\mu_X}$$
Now, say I also have random variable $Y$. Let us define:
$$ CV_{X,Y}= \sqrt{\frac{\text{Cov}(X,Y)}{\mu_X \mu_Y}} $$
So far I have found these properties of the latter:
- Dimensionless
- Nests $CV_X$ for $Y=X$
- Invariant to multiplicative transformations of one or both variables (like $CV$)
- Defined only when the term inside the squared root is positive, which discards e.g. negative covariance with positive means (unfortunately).
But, does such a unit make sense? I haven't found such measure around. Maybe it already has a name.
Update:
Replacing the covariance by the correlation yields:
$$ CV_{X,Y} = \sqrt{\rho_{X,Y}} \sqrt{CV_X} \sqrt{CV_Y} $$
If variables are perfectly correlated (1), then the "generalised" CV is the multiplication of square root of CVs. Interesting?
