Since the variance of the estimator for Pf corresponds to its confidence, variance reduction techniques aim at influencing the sampling such that the estimator variance becomes smaller, the confidence intervall narrower, hence the estimate of Pf becomes more accurate. A widely used technique is the Importance Sampling. The principle is to guide the sampling with any a priori information, such that the ratio of failure events to the total sample size increases. This of course is an intentional influence of the statistics, which has to be corrected to warrant unbiasedness of the estimator.
Samples are not generated following the prescribed density f X , but with a sampling density denoted as h Y. Setting h Y/h Y into the integral (Equation 5–14) does not change its value, but the integral can be interpreted as expected value within a population Y.
 | (5–18) |
The estimator now includes the correction term, also called importance sampling weight:
 | (5–19) |
This estimator can also be proved to be unbiased and consistent (Rubinstein 1981).