Response to Comment by Schilling

We appreciate the recent paper of Schilling and Stanley (2022, hereafter SS) on confidence intervals for the hypergeometric being brought to our attention, which we were not aware of while preparing our paper (Bartroff, Lorden, and Wang 2022, hereafter BLW) on that subject. Although there are commonalities between the two approaches, there are some important distinctions that we highlight here. Following those papers’ notations, below we denote the confidence intervals for the hypergeometric success parameter based on sample size n and population size N by LCO for SS, and C∗ for BLW. In the numerical examples below, LCO (github.com/mfschilling/ HGCIs) and C∗ (github.com/bartroff792/hyper) were computed using the respective authors’ publicly available R code, running on the same computer. Computational time. LCO and C∗ differ drastically in the amount of time required to compute them. Figure 1 shows the computational time of LCO and C∗ for α = 0.05, N = 200, 400, . . . , 1000, and n = N/2. For example, for N = 1000 the computational time of LCO exceeds 100 min whereas C∗ requires roughly 1/10th of a second (0.002 min). In further numerical comparisons not included here, we found this relationship to be common for moderate to large values of the sample and population sizes, n and N. This may be due to the algorithm for computing LCO which calls for searching among all acceptance functions of minimal span (SS, p. 37). Provable optimality. SS contains two proofs, one in the Appendix of a basic result about the hypergeometric parameters, and one in the main text of the paper’s only theorem (SS, p. 33) which is a general result that size-optimal hypergeometric acceptance sets are inverted to yield size-optimal confidence “intervals.” However, not all inverted acceptance sets will yield proper intervals, and in practice one often ends up with noninterval confidence sets, for example, intervals with “gaps.” This occurs when the endpoint sequences of the acceptance intervals being inverted are non-monotonic, or themselves have gaps. SS address this by modifying their proposal in this situation to mimic a method of Schilling and Doi (2014) developed for the Binomial distribution. SS (pp. 36–37) write, Where there is a need to resolve a gap, in which case the minimal span acceptance function that causes the gap is replaced with the one having the