Bartroff, J., Lorden, G. and Wang, L. (2022), “Optimal and Fast Confidence Intervals for Hypergeometric Successes,” The American Statistician: Comment by Schilling

Abstract

The article “Optimal and Fast Confidence Intervals for Hypergeometric Successes” by Bartroff, J., Lorden, G. and Wang, L. (BLW) develops a procedure for interval estimation of the number of successes M in a finite population based on constructing minimal length symmetrical acceptance intervals, which are inverted to determine confidence intervals based on the number of successes x obtained from a sample of size n. The authors compare their procedure to previously developed methods derived from the method of pivoting (Buonaccorsi 1987; Konijn 1973; Casella and Berger 2002, chap. 9) as well as to the more recent work of Wang (2015), and show that their approach generally leads to substantially shorter confidence intervals than those of these competitors, while frequently achieving higher coverage. However, the present authors BLW were evidently unaware of my recent paper with A. Stanley, “A New Approach to Precise Interval Estimation for the Parameters of the Hypergeometric Distribution” (Schilling and Stanley 2020), which solved the problem of constructing a hypergeometric confidence procedure that has minimal length (that is, minimal total cardinality of the confidence intervals for x = 0 to n), while maximizing coverage among all length minimizing procedures. We also compared our method to the same competitors as those listed above, as well as to one that can be obtained from Blaker’s (2000) method, and we showed the superiority in performance of our procedure. The two goals of our paper—length minimization and maximal coverage—are the same as those in BLW’s paper, and BLW’s approach matches rather closely with ours. The authors’ “α optimal” is our “minimal cardinality,” while our “maximal coverage” is BLW’s “PM-maximizing.” The only substantive difference between the two confidence procedures is that BLW’s specifies symmetrical acceptance sets, while ours does not. This affects only a small number of confidence intervals. An investigation of all 95% confidence intervals for each population size N between 5 and 100 and sample sizes n = 5, 10, . . . with n ≤ N finds that BLW’s confidence intervals are identical to ours in 99.43% of the 34,200 intervals checked. When they are different, the BLW