Importance Sampling in the Monte Carlo Study of Sequential Tests

Abstract

Let x1, x2,⋯ be independent random variables which under Pθ have probability density function of the form Pθ{xk ∈ dx} = exp(θ x - Ψ(θ)) dH(x), where Ψ is normalized so that Ψ(0) = Ψ'(0) = 0. Let $a \leqq 0 b, s_n = \sum^n_1 x_k$, and $T = \inf \{n: s_n ot\in (a, b)\}.$ For $u 0$, an unbiased Monte Carlo estimate of Pu(sT ≥ b) is the average of independent Pθ-realizations of I{sT ≥ 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.