Geodesic
On asymptotic efficiency of Kolmogorov--Smirnov-type tests for symmetry for random sample size
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Tome 153 (1986), pp. 122-128

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One considers Kolmogorov–Smirnov-type statistics and the sign statistic in the problem of the testing of symmetry for random sample size. Under some conditions one finds the large deviation asymptotics of these statistics and their Bahadur efficiency. 
@article{ZNSL_1986_153_a10,
     author = {Ya. Yu. Nikitin and B. Saidu},
     title = {On asymptotic efficiency of {Kolmogorov--Smirnov-type} tests for symmetry for random sample size},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {122--128},
     publisher = {mathdoc},
     volume = {153},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_153_a10/}
}
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Ya. Yu. Nikitin; B. Saidu. On asymptotic efficiency of Kolmogorov--Smirnov-type tests for symmetry for random sample size. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Tome 153 (1986), pp. 122-128. http://geodesic.mathdoc.fr/item/ZNSL_1986_153_a10/