In a recent article Bechhofer and Kulkarni proposed a class of closed adaptive sequential procedures for selecting that one of $k \geq 2$ Bernoulli populations with the largest single-trial success probability. These sequential procedures which take no more than $n$ observations from any one of the $k$ populations achieve the same probability of a correct selection as does a single-stage procedure which takes exactly $n$ observations from every one of the $k$ populations. In addition, they often require substantially less than a total of $kn$ observations to terminate sampling. Amongst other problems, Bechhofer and Kulkarni considered the problem of devising a procedure within this class which minimizes the expected total number of observations to terminate sampling. For their proposed procedure they cited several optimality properties for the case $k = 2$ and conjectured additional optimality properties for the case $k \geq 3$. In this article we use a new method of proof to establish stronger results than those cited by Bechhofer and Kulkarni for the case $k = 2$, and prove stronger results than those conjectured for $k \geq 3$. We also describe a new procedure for $k \geq 3$ and prove that it minimizes the expected total number of observations to terminate sampling when all of the success probabilities are small.
"Optimal Properties of the Bechhofer-Kulkarni Bernoulli Selection Procedure." Ann. Statist. 14 (1) 298 - 314, March, 1986. https://doi.org/10.1214/aos/1176349857