Randomised sequential probability ratio tests for stochastic processes

Abstract

For testing sequentially a simple hypothesis vs. a simple alternative, the likelihood ratio process ( r t ) does not have continuous sample paths in general. Consequently, (1) nominal errors α 0 and α 1 are almost never attained even though the total error is bounded by α 0+ α 1, and (2) the expected duration of observation ( ASN function) always exceeds the bound possible under the hypotheses. In the fixed sample size case, this problem is solved by randomisation at the boundaries. In this paper we propose a similar idea, viz., randomisation between two sequential probability ratio tests ( SPRT)—the usual one with boundaries, ( A, B) (A= log (α 1 (1−α 0) )<0 , B= log( (1−α 1) α 0) >0) and another SPRT with boundaries ( A′, B′), where A< A′<0< B′< B. We use the property that the Wiener process in which the log r t -process can be embedded has continuous sample paths.