Integral of CDF Difference as Comparison Measure

Question

Given two r.v. $X_1$ and $X_2$ taking values in $[0,2]$, and their CDFs (cumulative distribution functions) $F_1$ and $F_2$, I'm computing the following measure:

$M = \int_0^2 F_1(t) - F_2(t) dt$

From its sign and magnitude, this measure $M$ produces information that can intuitively be described as: what values are larger on average, values from $X_1$ or values from $X_2$.

I'm looking for some already known statistical notion/metric for a distribution comparison, that is related to $M$ in some way. Does $M$ has a mathematical name? Or is there any metric that has a tight connection to $M$?

Answer 1

Below all existing relations are described.

• The Kolmogorov-Smirnov statistic:

  $KS = \sup_t |F_1(t) - F_2(t)|$

  and the signed Kolmogorov-Smirnov statistic:

  $sKS = \sup_t F_1(t) - F_2(t)$

  are popular choices in the setting of two-sample test. They both have some similarities with $M$, yet there are also major differences (e.g. integral vs $\sup$).

• Likewise, the first Wasserstein distance between two densities can be shown to be equal to [Aaditya Ramdas et al.]:

  $W_1 = \int_{0}^{2} |F_1(t) - F_2(t)| dt$

  It is very close to $M$, but still has the absolute value.

• Finally, denoting by $\mu_1$ and $\mu_2$ the means of $X_1$ and $X_2$ respectively, and using the known relation between the expected value and CDF $\mu_i = \int_{0}^{2} 1 - F_i(t) dt$, we can conclude that:

  $M = \int_{0}^{2} F_1(t) - F_2(t) dt = \mu_2 - \mu_1$.

  Hence, $M$ is just a difference between the corresponding expected values.