Exact Optimal Confidence Intervals for Hypergeometric Parameters

For a hypergeometric distribution, denoted by Hyper(M,N,n), where <i>N</i> is the population size, <i>M</i> is the number of population units with some attribute, and <i>n</i> is the given sample size, there are two parametric cases: (i) <i>N</i> is unknown and <i>M</i> is given; (ii) <i>M</i> is unknown and <i>N</i> is given. For each case, we first show that the minimum coverage probability of commonly used approximate intervals is much smaller than the nominal level for any <i>n</i>, then we provide exact smallest lower and upper one-sided confidence intervals and an exact admissible two-sided confidence interval, a complete set of solutions, for each parameter. Supplementary materials for this article are available online.

对于记为Hyper(M,N,n)的超几何分布（hypergeometric distribution），其中N为总体容量，M为具有某一属性的总体单元数，n为给定的样本量，存在两类参数情形：(i) 总体容量N未知且已知具有该属性的总体单元数M；(ii) 具有该属性的总体单元数M未知且已知总体容量N。针对每一种情形，本文首先证明：对于任意样本量n，常用近似置信区间的最小覆盖概率远低于名义置信水平；随后针对每个参数，分别给出精确的最小单侧下限与上限置信区间，以及一套完整的精确可容许双侧置信区间解决方案。本文的补充材料可在线获取。