Truncated sequential hypothesis tests

Through a careful examination of the equations by which Wald determines the values of the boundaries for tests of sequential hypotheses, we are able to obtain interesting relationships for the conditional probability distributions of the stage at which the test terminates. These enable us to study sequential tests in which the boundaries are functions of the sample number. We are particularly concerned with tests with convergent boundaries, and investigate a set of boundaries which approach those of a truncated Wald test. Approximate expressions for the expected sample number and probabilities of error are obtained for the tests considered. The obtained approximations apply best for gently tapering slopes. Extensions of our method can be applied to evaluate various ad hoc schemes for truncating tests and to the theory of tests with a varying parameter.