Geodesic
Limit distributions and Bahadur efficiencies for Kolmogorov--Smirnov-type statistics with random indices
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 185-194

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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. 
@article{ZNSL_1976_55_a11,
     author = {Ya. Yu. Nikitin},
     title = {Limit distributions and {Bahadur} efficiencies for {Kolmogorov--Smirnov-type} statistics with random indices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {185--194},
     publisher = {mathdoc},
     volume = {55},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_55_a11/}
}
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Ya. Yu. Nikitin. Limit distributions and Bahadur efficiencies for Kolmogorov--Smirnov-type statistics with random indices. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 185-194. http://geodesic.mathdoc.fr/item/ZNSL_1976_55_a11/