The Mann-Whitney U Test can be used to test whether a randomly selected value from one distribution, $\ f_1(x)$, will be less than or greater than a randomly selected value from a second distribution, $\ f_2(x)$.
However when the two distributions are identical up to a simple translation, i.e. $\ f_1(x) = f_2(x + c)$, then a significant finding from this test implies that there exists a significant difference between the two populations' medians.
My question is, what's the best way to determine whether two populations are really identical up to translation? The most obvious thing that comes to mind is to translate the samples from the second distribution to bring it into alignment with the first, as determined by minimizing e.g. earth-mover distance, KL-divergence, or distance between medians, and then comparing the newly aligned distributions using the Kolmogorov-Smirnov test.
But I was wondering if there was a simpler one-step method, or some industry-standard methodology?