The Kolmogorov Smirnov test is based on the maximum vertical distance between the ECDFs of two provided samples.
Is there a variant that checks the maximum horizontal distance?
The Kolmogorov Smirnov test is based on the maximum vertical distance between the ECDFs of two provided samples.
Is there a variant that checks the maximum horizontal distance?
My guess as to why people don't use it: if you have a test that looks at horizontal distance it won't be distribution-free (at least not without some modification*).
Consider a continuous cdf and an ecdf plotted on the same axes with the KS-statistic marked in at the point of greatest vertical distance between them.
Note that a completely monotonic transformation of the x-axis will change the horizontal scale but leave the vertical distances unchanged. It's this feature that in essence makes the K-S test work the same for every fully-specified continuous distribution (since you can convert between them by exactly such a transformation).
But if you measure horizontal distance, every non-identity transform is going to change the distribution of horizontal distances.
* It might work if you convert everything back to standard uniform and then measure horizontal distance, but I don't think that's what you're asking about.