Extended Optimality of Sequential Probability Ratio Tests

The problem of sequentially testing two simple hypotheses for a stochastic process is considered. It is shown that, for arbitrary distributions P₀ and P₁, the following optimality holds for an SPRT which stops on its boundaries: If α and β represent the error probabilities of the SPRT and a competing test has error probabilities α' ≤ α and β' ≤ β then E₀g(Dτ') ≥ E₀g(D_τ) for any convex function g satisfying some minor requirement, provided P₁(τ' < ∞) = 1 for the competing test. Here D_τ and Dτ' denote the terminal likelihood ratios under the SPRT and the competitor. An analogous statement holds for expectation under P₁, and several applications of this optimality result are given.