Some observations on the distribution of order statistics under simple-random-sampling without replacement

This paper examines the distribution of order statistics from simple-random-sampling without replacement (SRSWOR) from a finite population composed of 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 allow for sampling from an arbitrary finite population which can include repeated values. Finally, 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.