On a non-parametric test for homogeneity of several samples

Let $(x_{i1},\dots,x_{in_i})$, $i=\overline{1,m}$ be independent samples of sizes $n_1,\dots,n_m$ from continuous distribution functions $F_1(x),\dots,F_m(x)$. For testing the hypothesis $H_0$: $F_1(x)=\dots=F_m(x)$, tests based on the statistics
$$ S(n_1,\dots,n_m)=\sup_{-\infty<x<\infty}\biggl(\sum_{i=1}^m c_i\biggl[F_{n_i}(x)-\biggl(\sum_{i=1}^m c_i F_{n_i}(x)\biggr)/\sum_{i=1}^m c_i\biggr]^2\biggr)^{1/2} $$
are considered where $F_{n_1}(x),\dots,F_{n_m}(x)$ are the empirical distribution functions of the samples and $c_1,\dots,c_m$ arbitrary positive numbers. Numerical methods for calculation of exact and limiting distributions of $S(n_1,\dots,n_m)$ under $H_0$ are described.