A Fast Simulation Method for the Sum of Subexponential Distributions

TL;DR

This paper introduces a new importance sampling framework for efficiently estimating the probability that the sum of heavy-tailed random variables exceeds a threshold, significantly reducing computational costs for rare event probabilities.

Contribution

It develops a minmax optimal importance sampling method based on hazard rate twisting for sums of heavy-tailed distributions, improving efficiency and asymptotic optimality.

Findings

The proposed IS method outperforms naive Monte Carlo in efficiency.

The approach achieves asymptotic optimality as thresholds grow large.

Numerical results confirm the near-optimality of the minmax parameter choice.

Abstract

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. Closed-form expression of the sum distribution is usually intractable and presents an open problem. A crude Monte Carlo (MC) simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive especially when dealing with rare events (i.e events with very small probabilities). Importance Sampling (IS) is an alternative approach which effectively improves the computational efficiency of the MC simulation. In this paper, we develop a general framework based on IS approach for the efficient estimation of the probability that the sum of independent and not necessarily identically distributed heavy-tailed RVs exceeds a given threshold. The proposed IS approach is based on constructing a new sampling distribution by twisting the hazard rate of the original underlying distribution of each component in the summation. A minmax approach is carried out for the determination of the twisting parameter, for any given threshold. Moreover, using this minmax optimal choice, the estimation of the probability of interest is shown to be asymptotically optimal as the threshold goes to infinity. We also offer some selected simulation results illustrating first the efficiency of the proposed IS approach compared to the naive MC simulation. The near-optimality of the minmax approach is then numerically analyzed.