On a statistic for testing the homogeneity of polynomial samples

We consider $M \geqslant 2$ independent polynomial samples with $N$ outcomes. For the case when $M$ and $N$ are fixed but sizes of samples tend to infinity we find limit distributions of a new statistic ${\sigma^2}$: chi-square distribution with $(M - 1)(N - 1)$ degrees of freedom if samples are statistically homogeneous, non-central chi-square distribution with the same number of degrees of freedom if samples are «convergent» to homogeneous ones, and normal distribution if samples are statistically nonhomogeneous.