Exact Optimal Confidence Intervals for Hypergeometric Parameters

Abstract

For a hypergeometric distribution, denoted by $\text{Hyper}(M,N,n)$, where N is the population size, M is the number of population units with some attribute, and n is the given sample size, there are two parametric cases: (i) N is unknown and M is given; (ii) M is unknown and N 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 n, 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.

• Capture-recapture sampling
• Coverage probability
• One-sided interval
• Set inclusion
• Two-sided interval.

Additional information

Notes on contributors

Weizhen Wang

Weizhen Wang is Visiting Professor, College of Applied Sciences, Beijing University of Technology, Beijing 100124, P. R. China. He is also Professor, Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435 (E-mail: [email protected]). The author thanks a referee, the associate editor, the co-editor, and Professor Daniel T. Voss for their helpful comments and suggestions that substantially improved the article and also thanks Yunyang E. Wang for editing the article.