Conditional frequentist sequential tests for the drift of Brownian motion

In this paper, we consider the problem of sequentially testing simple hypotheses concerning the drift of a Brownian motion process from a unified Bayesian and frequentist perspective. The conditional frequentist approach to testing statistical hypotheses has been utilized in a variety of settings to produce tests that are virtually equivalent to their objective Bayesian counterparts. Herein, we show that, at least for standard classes of stopping boundaries, the unified theory developed so far does not directly apply to the problem at hand. We thus motivate the need for a new conditioning strategy that ensures the existence of a conditional frequentist test whose answer essentially matches the objective Bayesian test. Under a quite general set of assumptions, we show that the new form of conditioning can still be interpreted as conditioning on the evidence present in the data as represented by $P$-values. Further properties of the resulting procedure, including ancillarity of the partition associated to the conditioning statistic and the characterization of the no-decision region, are studied in detail in the setting of the familiar sequential probability ratio test.