Rare-Event Simulation Without Variance Reduction: An Extreme Value Theory Approach

In estimating probabilities of rare events, crude Monte Carlo (MC) simulation is inefficient which motivates the use of variance reduction techniques. However, these latter schemes rely heavily on delicate analyses of underlying simulation models, which are not always easy or even possible. We propose the use of extreme value analysis, in particular the peak-over-threshold (POT) method which is popularly employed for extremal estimation of real datasets, in the simulation setting. More specifically, we view crude MC samples as data to fit on a generalized Pareto distribution. We test this idea on several numerical examples. The results show that our POT estimator appears more accurate than crude MC and, while crude MC can easily give a trivial probability estimate 0, POT outputs a non-trivial estimate with a roughly correct magnitude. Therefore, in the absence of efficient variance reduction schemes, POT appears to offer potential benefits to enhance crude MC estimates.

1 Introduction

A major goal of rare-event simulation is to estimate tiny probabilities that are triggered by rare but catastrophic events (Bucklew 2004; Juneja and Shahabuddin 2006; Rubino and Tuffin 2009). This problem has been of wide interest to various application areas such as queueing systems (Dupuis et al. 2007; Dupuis and Wang 2009; Blanchet et al. 2009; Blanchet and Lam 2014; Kroese and Nicola 1999; Ridder 2009; Sadowsky 1991; Szechtman and Glynn 2002), communication networks (Kesidis et al. 1993), finance (Glasserman 2003; Glasserman and Li 2005; Glasserman et al. 2008) and insurance (Asmussen 1985; Asmussen and Albrecher 2010). In recent years, with the extensive development of machine learning and artificial intelligence, rare-event simulation is also applied to evaluate the robustness of machine learning predictors (Webb et al. 2018; Bai et al. 2022) or quantify the risk of intelligent physical systems (Huang et al. 2017; O'Kelly et al. 2018; Zhao et al. 2016; Zhao et al. 2017; Arief et al. 2021). In using Monte Carlo (MC) to estimate rare-event probabilities, a main challenge is that, by its own nature, the target rare events seldom occur in the simulation experiments. Since sufficient hits on the target events are required to achieve meaningfully accurate estimation, this makes crude MC computationally costly as the required simulation size to attain enough accuracy becomes enormous.