*Bounty: 50*

*Bounty: 50*

I quite often find myself testing hypotheses in which the standard deviation of one (Normally distributed) variable is linked to (the mean of) another variable. I would like to be able to express the strength of this association by means of an index between [-1, 1], similar in spirit to a correlation coefficient. I feel like I can’t be the first one with this problem, so my first question is: does something like this exist? My second question is whether something I’ve come up with myself seems reasonable.

To express the problem more precisely, let $Z$ be a normally distributed variable:

$$

Z sim Nleft(0,sigma^2right)

$$

where the standard deviation $sigma$ is a linear function of some other variables:

$$

sigma=Xbeta+varepsilon

$$

where $X={x_1, x_2, …, x_p}$ is a set of predictor variables, and $beta$ is a vector of linear coefficients on these predictors. So compared to the familiar linear model, the difference is that we now have a linear prediction for the second, rather than the first moment of the distribution of $Z$.

Given some observations of $Z$ and $X$, we can find the maximum likelihood estimate of $beta$, which we’ll denote $hat{beta}$. Now the question is, how much of the ‘variance in variance’ of $Z$ is explained by this linear model? This leads me to the idea of using kurtosis. That is, because $Z$ is distributed as a mixture of Normals with different SDs and a common mean, it will be leptokurtic and thus have excess kurtosis w.r.t. a Normal distribution with constant variance. However, if we divided each observation of $Z$ by its SD (i.e. $dot{Z_i}=frac{Z_i}{sigma_i}$, where $sigma_i=X_ibeta$), we should be able to reduce its kurtosis (to the point where, if the changes in variance of $Z$ are perfectly predicted by our fitted model, we should be able to get rid of 100% of the excess kurtosis).

So the index I’m proposing (analogous to $R^2$) is:

$$

xi^2 = 1 – frac{left|text{Kurt}[Z/hat{sigma}]-3right|}{text{Kurt}[Z]-3}

$$

where $hat{sigma}=Xhat{beta}$. If our model explains no “variance in variance” at all, then the kurtosis should be just as high after we transform $Z$ as before, in which case $xi^2=0$. If we managed to explain away all the changes in variance, then ${Z}/{hat{sigma}}$ should be perfectly Normally distributed (with kurtosis of 3), and thus $xi^2=1$.

Does that seem reasonable? Did I just re-invent the wheel (or a dumb version of a wheel)?