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

Abstract

Let X 1 , X 2 , … be a discrete-time stochastic process with a distribution P θ , θ ∈ Θ , where Θ is an open subset of the real line. We consider the problem of testing a simple hypothesis H 0 : θ = θ 0 vs. a composite alternative H 1 : θ > θ 0 , where θ 0 ∈ 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, β ˙ 0 ( ψ , φ ) the derivative, at θ = θ 0 , 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 β ˙ 0 ( 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 P θ ) distribution of the process X 1 , X 2 , … . The structure of optimal sequential tests is characterized.