Open-Ended Tests for Koopman-Darmois Families

The generalized likelihood ratio is used to define a stopping rule for rejecting the null hypothesis θ = θ₀ in favor of θ > θ₀. Subject to a bound α on the probability of ever stopping in case θ = θ₀, the expected sample sizes for θ > θ₀ are minimized within a multiple of log log α⁻¹, the multiple depending on θ. 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.