Exact Computation of Minimum Sample Size for Estimation of Binomial Parameters

It is a common contention that it is an ``impossible mission’’ to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such a very old but also extremely important problem and demonstrate that the difficulty for obtaining the exact solution is not insurmountable. Unlike the classical approximate sample size method based on the central limit theorem, we develop a new approach for computing the minimum sample size that does not require any approximation. Moreover, our approach overcomes the conservatism of existing rigorous sample size methods derived from Bernoulli’s theorem or Chernoff bounds. Our computational machinery consists of two essential ingredients. First, we prove that the minimum of coverage probability with respect to a binomial parameter bounded in an interval is attained at a discrete set of finite many values of the binomial parameter. This allows for reducing infinite many evaluations of coverage probability to finite many evaluations. Second, a recursive bounding technique is developed to further improve the efficiency of computation.

💡 Research Summary

The paper tackles the long‑standing problem of determining the exact minimum sample size required to estimate a binomial proportion p within a prescribed absolute error ε and confidence level 1‑δ. While practitioners have traditionally relied on approximations derived from the Central Limit Theorem (CLT) or on conservative bounds such as Bernoulli’s inequality and Chernoff’s inequality, these methods either introduce uncontrolled approximation error or lead to overly large sample sizes. The authors propose a fully non‑approximate algorithm that guarantees the coverage probability Pr{|p̂ – p| < ε} ≥ 1‑δ for every possible value of p in a given interval