Efficient Importance Sampling for the Left Tail of Positive Gaussian Quadratic Forms

TL;DR

This paper introduces an importance sampling method with bounded relative error for efficiently estimating the very small probabilities in the left tail of quadratic forms in Gaussian vectors, outperforming naive Monte Carlo.

Contribution

The paper proposes a novel importance sampling estimator with bounded relative error for the left tail of Gaussian quadratic forms, improving efficiency over traditional methods.

Findings

Estimator achieves bounded relative error property.

Significantly reduces simulation runs needed for small probabilities.

Outperforms naive Monte Carlo and existing approximations in experiments.

Abstract

Estimating the left tail of quadratic forms in Gaussian random vectors is of major practical importance in many applications. In this letter, we propose an efficient importance sampling estimator that is endowed with the bounded relative error property. This property significantly reduces the number of simulation runs required by the proposed estimator compared to naive Monte Carlo (MC), especially when the probability of interest is very small. Selected simulation results are presented to illustrate the efficiency of our estimator compared to naive MC as well as some of the well-known approximations.