A Sharp Necessary Condition for Admissibility of Sequential Tests-- Necessary and Sufficient Conditions for Admissibility of SPRT'S

Abstract

Consider the problem of sequentially testing the hypothesis that the mean of a normal distribution of known variance is less than or equal to a given value versus the alternative that it is greater than the given value. Impose the linear combination loss function under which the risk becomes a constant $c$, times the expected sample size, plus the probability of error. It is known that all admissible tests must be monotone--that is, they stop and accept if $S_n$, the sample sum at stage $n$, satisfies $S_n \leq a_n$; stop and reject if $S_n \geq b_n$. In this paper we show that any admissible test must in addition satisfy $b_n - a_n \leq 2\bar{b}(c)$. The bound $2\bar{b}(c)$ is sharp in the sense that the test with stopping bounds $a_n \equiv -\bar{b}(c), b_n \equiv \bar{b}(c)$ is admissible. As a consequence of the above necessary condition for admissibility of a sequential test, it is possible to characterize all sequential probability ratio tests (SPRT's) regarding admissibility. In other words necessary and sufficient conditions for the admissibility of an SPRT are given. Furthermore, an explicit numerical upper bound for $\bar{b}(c)$ is provided.