I have two Gaussian random variables $X$ and $Y$, each of which is an estimator of an underlying quantity. I need to measure whether $Y$ is estimating something different than $X$. So if the mean of $Y$ is very different to the mean of $X$, that is good. However, if the variance of $Y$ is also much higher than the variance of $X$ it might just be that $Y$ is a poor estimator and the different mean is not meaningful (if you'll excuse the pun).
So for a fixed $X$, I need a function of $Y$ that increases with $\left|\mu_Y-\mu_X\right|$ but that decreases with $\sigma_Y$ (and is zero if and only if $\sigma_Y = \sigma_X$ and $\mu_Y = \mu_X$). I don't really know whether to call this a measure, a divergence or something else, since it will be negative when $\mu_Y = \mu_X$ and $\sigma_Y > \sigma_X$.
I thought that the Kullback-Leibler divergence might be the way to go but it responds the wrong way to changes in the variance of $Y$. Also, it can't be negative.
Clearly, something like $\left|\mu_Y-\mu_X\right| + \sigma_X - \sigma_Y$ has the above properties but I'm looking for something a little more grounded in information theory.
Ideally, this would generalize to distributions other than Gaussian.