This paper examines the distribution of order statistics taken from simple-random-sampling without replacement (SRSWOR) from a finite population with values 1,...,N. This distribution is a shifted version of the beta-binomial distribution, parameterised in a particular way. We derive the distribution and show how it relates to the distribution of order statistics under IID sampling from a uniform distribution over the unit interval. We examine properties of the distribution, including moments and asymptotic results. We also generalise the distribution to sampling without replacement of order statistics from an arbitrary finite population. We examine the properties of the order statistics for inference about an unknown population size (called the German tank problem) and we derive relevant estimation results based on observation of an arbitrary set of order statistics. We also introduce an algorithm that simulates sampling without replacement of order statistics from an arbitrary finite population without having to generate the entire sample.

翻译：本文研究了从具有值1,…,N的有限总体中进行无放回简单随机抽样（SRSWOR）所得顺序统计量的分布。该分布是贝塔-二项分布的一种特定参数化后的平移版本。我们推导了该分布，并展示了其与单位区间均匀分布下独立同分布抽样所得顺序统计量分布之间的关系。我们探讨了该分布的性质，包括矩和渐近结果。我们还将该分布推广到从任意有限总体中无放回抽样顺序统计量的情形。本文研究了顺序统计量在推断未知总体规模（即“德国坦克问题”）中的性质，并基于任意一组顺序统计量的观测结果推导了相关估计结果。此外，我们引入了一种算法，该算法可在无需生成整个样本的情况下，模拟从任意有限总体中无放回抽样顺序统计量的过程。