Importance Sampling in the Monte Carlo Study of Sequential Tests
The Annals of Statistics · 1976 · Volume 4 · Issue 4
Abstract
Let $x_1, x_2,\cdots$ be independent random variables which under $P_\theta$ have probability density function of the form $P_\theta\{x_k \in dx\} = \exp(\theta x - \Psi(\theta)) dH(x)$, where $\Psi$ is normalized so that $\Psi(0) = \Psi'(0) = 0.$ Let $a \leqq 0 < b, s_n = \sum^n_1 x_k$, and $T = \inf \{n: s_n \not\in (a, b)\}.$ For $u < 0$, an unbiased Monte Carlo estimate of $P_u(s_T \geqq b)$ is the average of independent $P_\theta$-realizations of $I_{\{s_T \geqq b\}} \exp\{(u - \theta)s_T - T(\Psi(u) - \Psi(\theta))\}$. It is shown that the choice $\theta = w$, where $w > 0$ is defined by $\Psi(w) = \Psi(u)$, is an asymptotically (as $b \rightarrow \infty)$ optimal choice of $\theta$ in a sense to be defined. Implications of this result for Monte Carlo studies in sequential analysis are discussed.
Keywords
62L10, 65C05, importance sampling, Monte Carlo, sequential test
Enable JavaScript for citation tools and additional paper interactions.