Conditions under which E(N1) = Infinity for Tong's Adaptive Solution to Ranking and Selection Problems

摘要

Tong proposed an adaptive approach as an alternative to the classical indifference-zone formulation of the problems of ranking and selection. With a fixed pre-selected gamma* (1/k < gamma* < 1) his procedure calls for the termination of vector-at-a-time sampling when the estimated probability of a correct selection exceeds gamma* for the first time. The purpose of this note is to show that for the case of two normal populations with common known variance, the expected number of vector-observations required by Tong's procedure to terminate sampling approaches infinity as the two population means approach equality for gamma* > or = approx. 0.8413. This phenomena presumably also persists if the two largest of k > or = 3 population means approach equality. Since in the typical ranking and selection setting it usually is assumed that that experimenter has no knowledge concerning the differences between the population means, the experimenter who uses Tong's procedure clearly does so at his own risk. (Author)