Geodesic
On the question of stability estimation in the problem of reconstructing the type of distribution
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 7-17

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The estimate for Levi distance between distribution functions of two types is received when the corresponding distributions of maximum invariant statistic $z=((x_1-\bar x)/s,\dots,(x_n-\bar x)/s)$ are close in some suitable sense. 
@article{ZNSL_1979_87_a1,
     author = {A. A. Zinger},
     title = {On the question of stability estimation in the problem of reconstructing the type of distribution},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {7--17},
     publisher = {mathdoc},
     volume = {87},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a1/}
}
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A. A. Zinger. On the question of stability estimation in the problem of reconstructing the type of distribution. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 7-17. http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a1/