  We study the problem of testing the goodness of fit of occurrences of items
from many categories to an identical Poisson distribution over the categories.
As a class of alternative hypotheses, we consider the removal of an $\ell_p$
ball, $p \leq 2$, of radius $\epsilon$ from a hypercube around the sequence of
uniform Poisson rates. When the expected number of samples $n$ and number of
categories $N$ go to infinity while $\epsilon$ is small, the minimax risk
asymptotes to $2\Phi(-n N^{2-2/p} \epsilon^2/\sqrt{8N})$; $\Phi(x)$ is the
normal CDF. This result allows the comparison of the many estimators previously
proposed for this problem at the constant level, rather than at the rate of
convergence of the risk or the scaling order of the sample complexity. The
minimax test relies exclusively on collisions in the small sample limit but
behaves like the chisquared test otherwise. Empirical studies over a range of
problem parameters show that the asymptotic risk estimate is accurate in finite
samples and that the minimax test is significantly better than the chisquared
test or a test that only uses collisions. Our analysis combines standard ideas
from non-parametric hypothesis testing with new results in the low count limit
of multiple Poisson distributions, including the convexity of certain kernels
and a central limit theorem of linear test statistics.
