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.