# zbMATH — the first resource for mathematics

A sharp necessary condition for admissibility of sequential tests - necessary and sufficient conditions for admissibility of SPRT’s. (English) Zbl 0539.62093
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).
Reviewer: R.A.Khan

##### MSC:
 62L10 Sequential statistical analysis 62C15 Admissibility in statistical decision theory 62L99 Sequential statistical methods
Full Text: