Asymptotic minimax nonparametric testing for independent sample density hypothesis

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 $\xi_N=\rho_NN^{2r/(4r+1)}$.