A power divergence test in the problem of sample homogeneity for a large number of outcomes and trials

In order to test homogeneity of $r$ independent polynomial schemes with the same number of outcomes $N$ under non-classical conditions where the numbers of trials $n_d$, $d=1,\dots,r$, in each of the schemes and the number of outcomes $N$ tend to infinity, we suggest a statistic $I(\lambda,r)$ which is a multidimensional analogue of the statistic $I(\lambda)$ introduced by T. Read and N. Cressie. We obtain conditions of asymptotic normality of the distributions of the statistics $I(\lambda)$ and $I(\lambda,r)$ for an arbitrary fixed integer $\lambda$, $\lambda\ne 0,-1$, as $N\to\infty$, $n_dN^{-1}\to\infty$, $d=1,\dots,r$. The expressions for the centring and normalising parameters are given in the explicit form for the hypothesis $H_0$ under which the distributions in these $r$ schemes coincide, and for some class of alternatives close to $H_0$.