An Essentially Complete Class of Decision Functions for Certain Standard Sequential Problems

A sequential problem is considered in which independent observations are taken on a chance variable X whose distribution can be represented by {equation*}{1}dG_θ(x) = ψ(θ)eθ x dμ(x),{equation*} where the parameter θ belongs to a given interval Ω of the real line but is otherwise unknown. The problem is to test H₁:θ θ^ against H₂:θ > θ^, where θ^ is a given point in Ω. Under certain assumptions the following class A is shown to be essentially complete relative to the class of decision rules with bounded risk functions. The decision rule δ ε A if and only if after taking n observations (i) δ depends on the observations only through n and vₙ = ∑ⁿi = 1 xᵢ and (ii) δ specifies a closed interval Jₙ: a₁ₙ, a₂ₙ for each n and the following rule of action (a) Stop experimentation as soon as vₙ ε Jₙ and (1) accept H₁ if vₙ < a₁ₙ (2) accept H₂ if vₙ > a₂ₙ. (b) If a₁ₙ < a₂ₙ take another observation if a₁ₙ < v < a₂ₙ. (c) If a₁ₙ < a₂ₙ and v = aᵢₙ, accept Hᵢ or take another observation or randomize between these two $(i = 1, 2)$. The Koopman-Darmois family of probability laws given above contains discrete members such as the binomial and Poisson distributions as well as absolutely continuous members such as the normal and exponential. It is interesting to note that the members of the class A can be obtained by starting with the sequential probability ratio test for testing some point θ^₁ θ^ against another point θ^₂ > θ^, namely, continue as long as B < ∏ⁿi = 1 ψ(θ^₂)eθ^₂ xᵢ∏ⁿi = 1 ψ(θ^₁)eθ^₁ xᵢ < A and replacing the constants $B, A$ by two arbitrary sequences Bₙ, Aₙ such that Bₙ Aₙ (n = 1, 2, ⋯).