A sharp necessary condition for admissibility of sequential tests - necessary and sufficient conditions for admissibility of SPRT's (Q794124)

Let \(X_ 1,X_ 2,..\). be i.i.d. \(N(\theta\),1) random variables, and consider the problem of sequentially testing \(H_ 0:\theta \leq 0\) versus \(H_ 1:\theta>0\). If N is a stopping rule of a sequential test, the underlying risk is \(cE_{\theta}N+P_{\theta}(error)\). Let \(S_ n=X_ 1+...+X_ n\), \(n\geq 1\), and let \(\{a_ n,b_ n\), \(n\geq 1\}\) be suitable sequences of constants. It is known that all admissible tests must be monotone in that N must be of the form \(N=\inf \{n\geq 1: S_ n\leq a_ n\) or \(S_ n\geq b_ n\}\), where \(H_ 0\) is accepted if \(S_ N\leq a_ N\), and rejected if \(S_ N\geq b_ N.\) The authors show that any admissible test must in addition satisfy \(b_ n-a_ n\leq 2\bar b(c).\) The bound is sharp in that the symmetric test with \(a_ n=-\bar b(c)\) and \(b_ n=\bar b(c)\) is admissible. As a consequence of the above result the authors obtain necessary and sufficient conditions for admissibility of an SPRT (sequential probability ratio test).