Optimal confidence interval for the difference of proportions

Estimating the probability of the binomial distribution is a basic problem, which appears in almost all introductory statistics courses and is performed frequently in various studies. In some cases, the parameter of interest is a difference between two probabilities, and the current work studies the construction of confidence intervals for this parameter when the sample size is small. Our goal is to find the shortest confidence intervals under the constraint of coverage probability being larger than a predetermined level. For the two-sample case, there is no known algorithm that achieves this goal, but different heuristics procedures have been suggested, and the present work aims at finding optimal confidence intervals. In the one-sample case, there is a known algorithm that finds optimal confidence intervals presented by Blyth and Still (1983). It is based on solving small and local optimization problems and then using an inversion step to find the global optimum solution. We show that this approach fails in the two-sample case and therefore, in order to find optimal confidence intervals, one needs to solve a global optimization problem, rather than small and local ones, which is computationally much harder. We present and discuss the suitable global optimization problem. Using the Gurobi package we find near-optimal solutions when the sample sizes are smaller than 15, and we compare these solutions to some existing methods, both approximate and exact. We find that the improvement in terms of lengths with respect to the best competitor varies between 1.5\% and 5\% for different parameters of the problem. Therefore, we recommend the use of the new confidence intervals when both sample sizes are smaller than 15. Tables of the confidence intervals are given in the Excel file in this link.