# Generalised coefficient of variation?

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?

• A further restriction is that neither mean can be zero. Also, you need the sign of covariance/product of means to be positive. So, negative covariance is compatible with one mean (only) being negative. These restrictions seem very artificial, so I don't hold hope for you that this measure is ever interesting or useful. – Nick Cox May 31 '18 at 9:53
• @NickCox Sorry, I made the implicit but unnecessary assumption that means are positive (they always are in the context I have in mind). Updated. – luchonacho May 31 '18 at 9:59