  Importance sampling (IS) is an important technique to reduce the estimation
variance in Monte Carlo simulations. In many practical problems, however, the
use of IS method may result in unbounded variance, and thus fail to provide
reliable estimates. To address the issue, we propose a method which can prevent
the risk of unbounded variance; the proposed method performs the standard IS
for the integral of interest in a region only in which the IS weight is bounded
and use the result as an approximation to the original integral. It can be
verified that the resulting estimator has a finite variance. Moreover, we also
provide a normality test based method to identify the region with bounded IS
weight (termed as the safe region) from the samples drawn from the standard IS
distribution. With numerical examples, we demonstrate that the proposed method
can yield rather reliable estimate when the standard IS fails, and it also
outperforms the defensive IS, a popular method to prevent unbounded variance.
