# Integral of Kolmogorov-Smirnov metrics?

I want to to calculate the area of the differences between the (red) null and (blue) empirical distributions of x, like in

, the arrow is the KS metric, but is there a name for the area between the curves? And how does it relate with Information Entropy (Shannon)?

If the distributions are not exactly identical, the area will be >0. Just to be clear, I am not interested by Area(red)-Area(blue), but by the integral |red-blue|. Not the difference of integrals, but the integral of absolute differences. The way I see it relating to entropy and information content is when you consider uniform distribution for the null.

Seb

• I don't know what that area is called, but it certainly is not a good metric for how different the null and empirical distribution are. The area could be 0 while both distributions are still very different. Note also that the difference in area is simply the difference in the expected value of both distributions. Commented Jan 8, 2018 at 13:13
• @StijnDeVuyst That is an accurate and (seemingly) complete reply. (Well, I suppose you could also point out the obvious: namely, that the difference in expectations has nothing at all to do with entropy.) If you get the time, please consider posting it as an answer.
– whuber
Commented Jan 8, 2018 at 14:24
• Thanks, there might be a misunderstanding then. If the distributions are not exactly identical, the area will be >0. Just to be clear, I am not interested by Area(red)-Area(blue), but by the integral |red-blue|. Not the difference of integrals, but the integral of absolute differences. The way I see it relating to entropy and information content is when you consider uniform distribution for the null. Commented Jan 8, 2018 at 14:31
• You should make this clarifications as an edit to the post! Comments are easily overlooked, and can be deleted. Commented Jul 8, 2022 at 3:18
• The signed area will equal the difference in expected values. The integral of the absolute difference is, by definition, the $L^1$ distance between the two distribution functions.
– whuber
Commented Aug 3, 2022 at 14:42