Bahadur efficiency of $\omega^2$-type criteria in the several sample case

We consider the problem of testing of the hypothesie that $r$ independent samples of sizes $n_1,n_2,\dots,n_r$, are drawn from the some population with continuous distribution function $F$. We obtain the local exact slope in the Bahadur sense of the statistic
$$ \omega^k_{n_1,n_2,\dots,n_r;q}=\sum_{j=1}^r\rho_j^{k/3} \int_{-\infty}^\infty[F_{n_j}^{(j)}(t)-F(t)]^kq(F(t))\,dF(t), $$
where $F_{n_j}^{(j)}(t)$ are ampirical distribution functions, $q$ is a weight function, $k$ a natural number.