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：本文研究了来自有限总体（取值为1,...,N）的简单随机无放回抽样（SRSWOR）中顺序统计量的分布。该分布是贝塔-二项分布的一种特殊参数化平移形式。我们推导了这一分布，并展示了其与单位区间上均匀分布独立同分布抽样中顺序统计量分布的联系。我们分析了该分布的性质，包括矩和渐近结果。进一步，我们将该分布推广至任意有限总体中无放回抽样的顺序统计量。针对未知总体规模的推断问题（即德国坦克问题），我们研究了顺序统计量的相关性质，并基于任意顺序统计量集合的观测值推导了相应的估计结果。此外，我们还提出了一种算法，该算法无需生成完整样本即可模拟任意有限总体中无放回抽样的顺序统计量。