Two-sample tests based on the integrated empirical process

Let X$_1$, …, X$_m$ and Y$_1$, …, Y$_n$ be independent random variables, where X$_1$, …, X$_m$ are i.i.d. with continuous distribution function (df) F, and Y$_1$, …, Y$_n$ are i.i.d. with continuous df G. For testing the hypothesis H$_0$: F = G, we introduce and study analogues of the celebrated Kolmogorov–Smirnov and one- and two-sided Cramér-von Mises statistics that are functionals of a suitably integrated two-sample empirical process. Furthermore, we characterize those distributions for which the new tests are locally Bahadur optimal within the setting of shift alternatives.