Locally Most Powerful Sequential Tests of a Simple Hypothesis vs One-Sided Alternatives

Let X1,X2, . . . be a discrete-time stochastic process with a distribution Pθ, θ ∈ Θ, where Θ is an open subset of the real line. We consider the problem of testing a simple hypothesis H0 : θ = θ0 versus a composite alternative H1 : θ > θ0, where θ0 ∈ Θ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test (ψ, φ) with a (randomized) stopping rule ψ and a (randomized) decision rule φ let α(ψ, φ) be the type I error probability, β̇0(ψ, φ) the derivative, at θ = θ0, of the power function, and N (ψ) an average sample number of the test (ψ, φ). Then we are concerned with the problem of maximizing β̇0(ψ, φ) in the class of all sequential tests such that α(ψ, φ) ≤ α and N (ψ) ≤ N , where α ∈ [0, 1] and N ≥ 1 are some restrictions. It is supposed that N (ψ) is calculated under some fixed (not necessarily coinciding with one of Pθ) distribution of the process X1,X2 . . . . The structure of optimal sequential tests is characterized.