Exact computation of minimum sample size for estimation of binomial parameters

摘要

It is a common contention that it is an ``impossible mission'' to exactlydetermine the minimum sample size for the estimation of a binomial parameterwith prescribed margin of error and confidence level. In this paper, weinvestigate such a very old but also extremely important problem anddemonstrate that the difficulty for obtaining the exact solution is notinsurmountable. Unlike the classical approximate sample size method based onthe central limit theorem, we develop a new approach for computing the minimumsample size that does not require any approximation. Moreover, our approachovercomes the conservatism of existing rigorous sample size methods derivedfrom Bernoulli's theorem or Chernoff bounds. Our computational machinery consists of two essential ingredients. First, weprove that the minimum of coverage probability with respect to a binomialparameter bounded in an interval is attained at a discrete set of finite manyvalues of the binomial parameter. This allows for reducing infinite manyevaluations of coverage probability to finite many evaluations. Second, arecursive bounding technique is developed to further improve the efficiency ofcomputation.