Resumen
In the sequential analysis of data, both Bayesian and frequentist methods often make use of stopping rules or stopping boundaries. For example, repeated significance tests boundaries, and other boundaries, are being used in clinical trials more and more often. Lerche [1986a. An optimal property of the repeated significance test. Proc. Nat. Acad. Sci. 83, 1546-1548] studies a specific problem based on Brownian motion with unknown drift @q. The problem is to test H"0:@q<0 against H"1:@q>0. Lerche shows that a repeated significance test boundary based on frequentist principles is also an optimal Bayesian boundary when the decision loss is 0-1 and the sampling cost per unit time is c@q^2 with c>0. In this paper we look at extensions of this idea. For the null and alternative hypotheses listed above, and under Lerche's sampling cost c@q^2, we show that there exists a family of loss functions such that a given boundary, under some restrictions, is Bayes. While the sampling cost is as simple as Lerche's, the loss function is too complicated to be useful. It also has the atypical property of depending on both the stopping time and the value of the Brownian motion upon stopping. Therefore, suboptimal procedures with simple loss/cost structures are also developed.