## The Annals of Statistics

### Importance Sampling in the Monte Carlo Study of Sequential Tests

D. Siegmund

#### 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.

#### Article information

Source
Ann. Statist. Volume 4, Number 4 (1976), 673-684.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176343541

Digital Object Identifier
doi:10.1214/aos/1176343541

Mathematical Reviews number (MathSciNet)
MR418369

Zentralblatt MATH identifier
0353.62044

JSTOR