Sequential probability ratio tests based on improper priors

Abstract

One way of handling composite hypotheses by SPRT's is to integrate out nuisance parameters under both the null and alternative hypotheses and then form an SPRT. If the prior is improper, Wald inequalities will not hold in general. We provide sufficient conditions and then verify that the usual sequential |t|-test satisfies these conditions but Wald's |t|-test does not satisfy these conditions and in fact cannot satisfy a strengthened form of these inequalities in general. Nonetheless, simulations show Wald's inequalities for A, B, corresponding to usual small α, β, will usually hold.