Some generalizations of the empty boxes test

Let us suppose that $n$ balls are distributed among $N$ boxes so that each ball may $N$ fall into the ith box with probability $a_i$ ($a_i\ge0$, $\sum_{i=1}^Na_i=1$) independently of what happens to the other balls. Let $\mu_r$ denote the number of boxes in which we have exactly $r$ balls. There are two hypotheses about $a_i$, $i=1,\dots,N$ approaching each other as $N$ increases. To distinguish these hypotheses statistical tests based on $\mu_0,\mu_1,\dots,\mu_r$ are considered. The most powerful test among the ones based on the linear statistics $\xi_r=c_{0r}\mu_0+\dots+c_{rr}\mu_r$ is found. This test is proved to coincide asymptotically with the Neyman–Pearson test e.g. it is the optimal one in the class of all the tests based on $\mu_0,\mu_1,\dots,\mu_r$.