Geodesic
Asymptotic minimax nonparametric testing for independent sample density hypothesis
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part VI, Tome 136 (1984), pp. 74-96

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The paper investigates the condition of minimax discemability for statistical hypothesis about sample of length $N\to\infty$ from interval $[0; 1]$ as function of asymptotic distance $\rho_N$ in $L_2[0;1]$ between sets of densities, which are conform to hypothesis and alternative, and densities degree $r$ of smoothness in $L_2[0;1]$: it is shown that defining value is $\xi_N=\rho_NN^{2r/(4r+1)}$. 
@article{ZNSL_1984_136_a5,
     author = {Yu. I. Ingster},
     title = {Asymptotic minimax nonparametric testing for independent sample density hypothesis},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {74--96},
     publisher = {mathdoc},
     volume = {136},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_136_a5/}
}
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Yu. I. Ingster. Asymptotic minimax nonparametric testing for independent sample density hypothesis. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part VI, Tome 136 (1984), pp. 74-96. http://geodesic.mathdoc.fr/item/ZNSL_1984_136_a5/