On the absolute significance test for polynomial distribution

Upper estimates are found for the sum of probabilities of all the events $(x_1\ldots x_r)$, where $x_k$ is the frequency of the $k$th outcome in $n$ independent trials carried out according to a polynomial scheme of trials with $r$ possible outcomes, the probability of each of which does not exceed the probability of a fixed event observed in $n$ independent trials carried out according to the same scheme. Using these estimates we construct a test rejecting a polynomial scheme when the probabilities of outcomes in it are known.