Geodesic
Large deviations and asymptotic efficiency of integral type statistics.~II
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 151-175

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The paper contains some results on asymptotic behaviour of large deviation probabilities for integral statistics $\int_0^1(F_n(t)-t)^kq(t)\,dt$ and their two-sample analogues. These results permits to calculate local exact Bahadur efficiency of statistics under consideration. 
@article{ZNSL_1980_97_a14,
     author = {Ya. Yu. Nikitin},
     title = {Large deviations and asymptotic efficiency of integral type {statistics.~II}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--175},
     publisher = {mathdoc},
     volume = {97},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a14/}
}
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Ya. Yu. Nikitin. Large deviations and asymptotic efficiency of integral type statistics.~II. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 151-175. http://geodesic.mathdoc.fr/item/ZNSL_1980_97_a14/