Error Probabilities and Average Sample Number of the Sequential Probability Ratio Test

Summary Let x 1, x 2,. . . be independent random variables which under p θ have probability density function of the form exp (θx – Ψ (θ) } relative to some measure v, where Ψ is normalized so that Ψ (0) = Ψ‘ (0) = 0. Let a ≤ 0 b, sn = ∑1 n x k, and T=infn: sn∉ (a, b). Approximations are given for p θsT ≥; b and E θ (T) based on asymptotic considerations arising in renewal theory. The accuracy of these approximations is discussed for normally distributed x‘s. It is shown that for θ = 0 these approximations give complete asymptotic expansions as b → ∞ in powers of b −1 with exponentially small remainder.