Project Euclid

The generalized likelihood ratio is used to define a stopping rule for rejecting the null hypothesis $\theta = \theta_0$ in favor of $\theta > \theta_0$. Subject to a bound $\alpha$ on the probability of ever stopping in case $\theta = \theta_0$, the expected sample sizes for $\theta > \theta_0$ are minimized within a multiple of $\log \log \alpha^{-1}$, the multiple depending on $\theta$. An heuristic bound on the error probability of a likelihood ratio procedure is derived and verified in the case of a normal mean by consideration of a Wiener process. Useful lower bounds on the small-sample efficiency in the normal case are thereby obtained.