We consider the problem of testing the fit of a discrete 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 $p \leq 2$. We deliver a sharp characterization of the asymptotic minimax risk when $\epsilon \to 0$ as the number of samples and number of dimensions go to infinity, for testing based on the occurrences' histogram (number of absent categories, singletons, collisions, ...). For example, for $p=1$ and in the limit of a small expected number of samples $n$ compared to the number of categories $N$ (aka "sub-linear" regime), the minimax risk $R^*_\epsilon$ asymptotes to $2 \bar{\Phi}\left(n \epsilon^2/\sqrt{8N}\right) $, with $\bar{\Phi}(x)$ the normal survival function. 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 is based on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting to a Bayesian one, and the reduction of a multi-dimensional optimization problem to a one-dimensional problem.

翻译：我们考虑从多个类别中抽取的离散样本与均匀分布拟合检验问题。作为备择假设类，我们考虑移除均匀率序列周围半径为$\epsilon$的$\ell_p$球（$p \leq 2$）。当$\epsilon \to 0$且样本数与维度数趋于无穷时，针对基于出现频次直方图（缺失类别数、独现数、碰撞数等）的检验，我们给出了渐近极小化极大风险的精确刻画。例如，对于$p=1$且样本期望数$n$远小于类别数$N$（即"亚线性"区间），极小化极大风险$R^*_\epsilon$渐近趋于$2 \bar{\Phi}\left(n \epsilon^2/\sqrt{8N}\right)$，其中$\bar{\Phi}(x)$为正态生存函数。对一系列问题参数的实证研究表明，该估计在有限样本下精确有效，且我们提出的检验显著优于卡方检验或仅使用碰撞的检验。我们的分析基于直方图坐标的渐近正态性、极小化极大框架与贝叶斯框架的等价性，以及将多维优化问题降维为一维问题的方法。