Optimal allocation in sequential tests comparing the means of two Gaussian populations

Abstract

SUMMARY The invariant sequential probability ratio test used in testing for a difference between the means of two Gaussian populations is set up. The error probabilities for this test are effectively constant over a rich class of data-dependent allocation rules. The additional risk, average sample number plus (y - 1) times the expected number of observations to the inferior population, for y > 1, is introduced and the optimal allocation rule is found for the continuous-time analogue to this problem. Analytical results show this rule to be asymptotically optimal in discrete time, and simulations indicate its near optimal per- formance for the finite case. The problem of two-population hypothesis testing with data-dependent allocation of observations has been treated by several authors. Also, the applications of this decision model, especially to clinical testing, have been well documented. Recent results show that when the test is sequential and the termination rule is of the sequential probability ratio test type, the probability of correct hypothesis selection is constant, ignoring overshoot, for a rich class of data-dependent allocation rules; see Flehinger, Louis, Robbins & Singer (1972) and an as yet unpublished paper of mine. This constancy permits one to search the class for a rule which performs well with respect to some additional cost structure, one usually based on the number of observations taken on the superior and inferior populations. Flehinger & Louis (1972) and Robbins & Siegmund (1974) give simulations showing that a substantial reduction in the expected number of observations on the inferior popu- lation is possible using data-dependent allocation rules, as opposed to equal assignment, for the case of comparing two Gaussian populations with known variances. The simulation results of Flehinger & Louis (1971) show the same reduction for the exponential distri- bution. In the present paper the risk function formed from the average sample number plus (y - 1) x the expected number of observations allocated to the inferior population, for y > 1, is introduced into the Gaussian testing model. Here y is the relative cost of taking an observation from the inferior as opposed to the superior population, and varying y allows one to balance the two components of risk. In ? 2 first the Gaussian allocation and testing problem is set up and previous results are summarized. Using the above risk function, in Appendix A the optimal allocation and its risk are obtained for the continuous-time idealization, that of comparing the drifts of two Brownian motions. Back in ? 2 this optimal rule is related to the discrete testing situation.