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

Abstract

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.

Recommended by N. Mukhopadhyay

Keywords:

• Adaptive allocation
• Multiple hypothesis testing
• Sampling cost
• Sequential selection

Subject Classifications:

• 62F07
• 62L10

ACKNOWLEDGMENTS

The authors want to express their gratitude to the five referees and the associate editor for their valuable comments, which led to substantial improvement of the original manuscript. Chan's research was supported by the National University of Singapore, and Lai's research was supported by the National Science Foundation.

Notes

Recommended by N. Mukhopadhyay