Nearly Optimal Sequential Tests of Composite Hypotheses

A simple class of sequential tests is proposed for testing the one-sided composite hypotheses H₀: θ ≤ θ₀ versus H₁: θ ≥ θ₁ for the natural parameter θ of an exponential family of distributions under the 0-1 loss and cost c per observation. Setting θ₁ = θ₀ in these tests also leads to simple sequential tests for the hypotheses H: θ < θ₀ versus K: θ > θ₀ without assuming an indifference zone. Our analytic and numerical results show that these tests have nearly optimal frequentist properties and also provide approximate Bayes solutions with respect to a large class of priors. In addition, our method gives a unified approach to the testing problems of H versus K and also of H₀ versus H₁ and unifies the different asymptotic theories of Chernoff and Schwarz for these two problems.