Importance Sampling in the Monte Carlo Study of Sequential Tests

Let x₁, x₂,⋯ be independent random variables which under P_θ have probability density function of the form P_θₖ ∈ dx\ = exp(θ x - Ψ(θ)) dH(x), where Ψ is normalized so that Ψ(0) = Ψ'(0) = 0. Let a 0 < b, sₙ = ∑ⁿ₁ xₖ, and T = inf : sₙ ∈ (a, b)\. For $u < 0$, an unbiased Monte Carlo estimate of Pᵤ(sT b) is the average of independent P_θ-realizations of I_ b\ exp\(u - θ)sT - T(Ψ(u) - Ψ(θ))\. It is shown that the choice θ = w, where $w > 0$ is defined by Ψ(w) = Ψ(u), is an asymptotically (as b → ∞) optimal choice of θ in a sense to be defined. Implications of this result for Monte Carlo studies in sequential analysis are discussed.