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Estimating the left tail of quadratic forms in Gaussian random vectors is ofmajor practical importance in many applications. In this paper, we propose anefficient and robust importance sampling estimator that is endowed with thebounded relative error property. This property significantly reduces the numberof simulation runs required by the proposed estimator compared to naive MonteCarlo. Thus, our importance sampling estimator is especially useful when theprobability of interest is very small. Selected simulation results arepresented to illustrate the efficiency of our estimator compared to naive MonteCarlo in both central and non-central cases, as well as both real and complexsettings.

Eficient Monte Carlo Simulation of the Left Tail of Positive Gaussian Quadratic Forms