Sequential Generalized Likelihood Ratios and Adaptive Treatment Allocation for Optimal Sequential Selection

Given k ≥ 2 populations from an exponential family, we consider herein the problem of efficient sequential selection of the population with the largest mean subject to a correct selection probability constraint. The selection procedure consists of a sampling rule, a stopping rule, and a terminal decision rule. Efficiency at every parameter configuration is measured by the expected total sampling cost together with the correct selection probability. By using sequential generalized likelihood ratio tests of multiple hypotheses and an adaptive sampling rule based on a constrained optimization problem, we show that it is possible to achieve asymptotic efficiency at the true (but unknown) parameter configuration as the probability of incorrect selection approaches 0, thereby resolving a number of open problems in this area. Finite-sample efficiency of the proposed procedure is demonstrated in simulation studies that also compare the procedure with other sequential selection procedures in the literature.