Limit distributions and Bahadur efficiencies for Kolmogorov–Smirnov-type statistics with random indices

One obtains limit distributions for some statistics with random indice of the Kolmogorov–Smirnov type considering the weak convergence of the corresponding empirical process. Approximate and exact Bahadur efficiencies of these statistics are computed. It is shown that they are in some sense worse that classical Kolmogorov–Smirnov statistics.