The minimax risk in testing the histogram of discrete distributions for uniformity under missing ball alternatives

We consider the problem of testing the fit of a sample of items from many categories to the uniform distribution over the categories. As a class of alternative hypotheses, we consider the removal of an $\ell_p$ ball of radius $\epsilon$ around the uniform rate sequence for $p \leq 2$. When the number of samples $n$ and number of categories $N$ go to infinity while $\epsilon$ goes to zero, the minimax risk $R_\epsilon^*$ in testing based on the sample's histogram (number of absent categories, singletons, collisions, ...) asymptotes to $2\Phi(-n N^{2-2/p} \epsilon^2/\sqrt{8N})$, with $\Phi(x)$ the normal CDF. This characterization allows comparing the many estimators previously proposed for this problem at the constant level rather than the rate of convergence of their risks. The minimax test mostly relies on collisions when $n/N$ is small, but otherwise behaves like the chisquared test. Empirical studies over a range of problem parameters show that this estimate is accurate in finite samples and that our test is significantly better than the chisquared test or a test that only uses collisions. Our analysis relies on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting and a Bayesian setting, and the characterization of the least favorable prior by reducing a multi-dimensional optimization problem to a one-dimensional problem.