Scale Invariant Statistical Distances Problem
Suppose we have empirical distributions to two $n$-dimensional random variables $X = (X_1, X_2, ..., X_n)$ and $Y = (Y_1, Y_2, ..., Y_n)$. The goal is to find $k < n$ components, such that the $1$-dimensional distributions of $X_{i_1}, X_{i_2}, ..., X_{i_k}$ and $Y_{i_1}, Y_{i_2}, ..., Y_{i_k}$ are "most similar" to each other, i.e. $\text{argmin}_i dist(X_i, Y_i)$.
We compare $X_1$ to $Y_1$, $X_2$ to $Y_2$ etc.
As an added difficulty due to physical units, the scale (max, min, std) of $X_i$ might be very differnt from the scale of $X_j$ for any $i \neq j$. But $X_i$ and $Y_i$ are approximately comparable (think same measurement device and test case but slightly different product).
Question: What statistical distances are available that capture similarity between the components of $X$ and $Y$, are scale invariant and emphasize tail probabilities?
Our current aproach is as follows:
For each $i=1,...,n$

*

*Take the Union $U_i$ over all samples from $X_i$ and $Y_i$

*Compute $m = mean(U_i), s = std(U_i)$

*$X_i' = (X_i - m) / s, \, Y_i‘ = (Y_i - m) / s$

*$d(X_i, Y_i) = \int_{\mathbb{R}} |p_i(x) - q_i(x)| |x|^p dx$ where $p$ denotes the PDF of $X‘_i$ and  and $q_i$ denotes the PDF of $Y‘_i$.

The factor $|x|^p$, ($p=5$) stems from the fact, that we don't really care about density around $0$.

There's probably a better answer incoming but here's something to get you started. It looks like what you're using is related to the Total Variation Distance
$$
\delta(P,Q) = \frac12 \int_{\mathbb{R}}|P(x) - Q(x)|\,dx
$$
(see this question for why this is the correct formula). This is unit independent since $P(x)$ is in units of $\frac{1}{x}$ and $dx$ is in units of $x$. It's also bounded between 0 and 1, making it a reasonable things to use to compare distribution over spaces with different units. The addition of the $x^p$ however will add in units and make comparison between the components $i$ and $j$ difficult.
If you want to keep the $x^p$ you could form a sort of "almost" $L^p$ metric using
$$
\delta(P,Q) = \left(\int_{\mathbb{R}}|P(x) - Q(x)|^px^{p-1}\,dx\right)^{\frac1p}
$$
This will tend to emphasize values away from 0 more, but I would hesitate to say more about the behavior of the metric, or how to metric will change for different distribution, you could do some computational investigations though.
You can find a long list of metrics here but no others jumped out at me as scale invariant.
