Lower Bounds for the Expected Sample Size and the Average Risk of a Sequential Procedure

Abstract

Sections 1-6 are concerned with lower bounds for the expected sample size, $E_0(N)$, of an arbitrary sequential test whose error probabilities at two parameter points, $\theta_1$ and $\theta_2$, do not exceed given numbers, $\alpha_1$ and $\alpha_2$, where $E_0(N)$ is evaluated at a third parameter point, $\theta_0$. The bounds in (1.3) and (1.4) are shown to be attainable or nearly attainable in certain cases where $\theta_0$ lies between $\theta_1$ and $\theta_2$. In Section 7 lower bounds for the average risk of a general sequential procedure are obtained. In Section 8 these bounds are used to derive further lower bounds for $E_0(N)$ which in general are better than (1.3).