Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order $|S|^\frac{1}{2}$. On the other hand, if $S=\{1,2,\ldots,|S|\}$, the maximum of the sample blown up by $n/(n-1)$ gives an efficient estimator based on any growing sample size. This paper gives refinements that interpolate between these extremes. A general non-asymptotic theory is developed. This includes estimating the volume of a compact convex set, the unseen species problem, and a host of testing problems that follow from the question `Is this new observation a typical pick from a large prespecified population?' We also treat regression style predictors. A general theorem gives non-parametric finite $n$ error bounds in all cases.

翻译：设 $S$ 为一有限集，$X_1,\ldots,X_n$ 是从 $S$ 中独立同分布均匀抽取的样本。在缺乏额外结构的情况下，为估计规模 $|S|$，可通过等待重复出现并利用生日问题来实现，此时所需样本量级为 $|S|^\frac{1}{2}$。另一方面，若 $S=\{1,2,\ldots,|S|\}$，则经因子 $n/(n-1)$ 放大的样本最大值可基于任意递增样本量给出有效估计。本文提出介于上述两种极端情形之间的精细化改进方法，并建立了一般的非渐近理论。该理论涵盖：紧凸集体积估计、未知物种问题，以及源于"新观测是否来自大型预设总体的典型抽取"这一问题的系列检验问题。我们还处理了回归型预测器，并给出一个适用于所有情形的通用定理，该定理提供了非参数有限样本误差界。