Locally most powerful sequential tests of a simple hypothesis vs. one-sided alternatives

Abstract

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 vs. 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, over(β, ̇)0 (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 over(β, ̇)0 (ψ, φ) in the class of all sequential tests such thatα (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. © 2009 Elsevier B.V. All rights reserved.