The properties of the generalized Waring distribution defined on the non negative integers are reviewed. Formulas for its moments and its mode are given. A construction as a mixture of negative binomial distributions is also presented. Then we turn to the Petersen model for estimating the population size $N$ in a two-way capture recapture experiment. We construct a Bayesian model for $N$ by combining a Waring prior with the hypergeometric distribution for the number of units caught twice in the experiment. Confidence intervals for $N$ are obtained using quantiles of the posterior, a generalized Waring distribution. The standard confidence interval for the population size constructed using the asymptotic variance of Petersen estimator and .5 logit transformed interval are shown to be special cases of the generalized Waring confidence interval. The true coverage of this interval is shown to be bigger than or equal to its nominal converage in small populations, regardless of the capture probabilities. In addition, its length is substantially smaller than that of the .5 logit transformed interval. Thus a generalized Waring confidence interval appears to be the best way to quantify the uncertainty of the Petersen estimator for populations size.

翻译：回顾了定义在非负整数上的广义Waring分布的性质，给出了其矩和众数的公式，并提出了其作为负二项分布混合的构造方法。随后，我们转向用于估计双路捕获-再捕获实验中总体规模$N$的Petersen模型。通过将Waring先验与实验中两次捕获单元数量的超几何分布相结合，我们构建了关于$N$的贝叶斯模型。利用后验分布（一种广义Waring分布）的分位数，得到了$N$的置信区间。研究表明，基于Petersen估计量的渐近方差和0.5 logit变换区间构建的总体规模标准置信区间是广义Waring置信区间的特例。在总体较小的情况下，无论捕获概率如何，该区间的真实覆盖率都大于或等于其名义覆盖率。此外，其长度明显小于0.5 logit变换区间。因此，广义Waring置信区间似乎是量化Petersen估计量关于总体规模不确定性的最佳方法。