Two-sample Kolmogorov-Smirnov test ($D_{n,m}={\sup }_x|F_n(x)-G_m(x)|$)

The two-sample Kolmogorov-Smirnov test compares two empirical distribution functions from independent samples, each internally independent and identically distributed, under the null hypothesis that their underlying probability distributions coincide. Both curves are estimated from data, so calibration depends on both sample sizes; the usual asymptotic scaling is $\sqrt{nm/(n+m)}{D}_{n,m}$. Tied data can be handled by an appropriate permutation test under the required exchangeability, rather than by assuming the untied continuous-null law.^[ref]