Efficient Importance Sampling for Large Sums of Independent and Identically Distributed Random Variables

by   Nadhir Ben Rached, et al.

We aim to estimate the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., ℙ(∑_i=1^NX_i≤γ), via importance sampling (IS). We are particularly interested in the rare event regime when N is large and/or γ is small. The exponential twisting is a popular technique that, in most of the cases, compares favorably to existing estimators. However, it has several limitations: i) it assumes the knowledge of the moment generating function of X_i and ii) sampling under the new measure is not straightforward and might be expensive. The aim of this work is to propose an alternative change of measure that yields, in the rare event regime corresponding to large N and/or small γ, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. For distributions whose probability density functions (PDFs) are 𝒪(x^d), as x → 0 and d>-1, we prove that the Gamma IS PDF with appropriately chosen parameters retrieves asymptotically, in the rare event regime, the same performance of the estimator based on the use of the exponential twisting technique. Moreover, in the Log-normal setting, where the PDF at zero vanishes faster than any polynomial, we numerically show that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting change of measure. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large N and/or small γ.


page 1

page 2

page 3

page 4


Rare tail approximation using asymptotics and L^1 polar coordinates

In this work, we propose a class of importance sampling (IS) estimators ...

A Koopman framework for rare event simulation in stochastic differential equations

We exploit the relationship between the stochastic Koopman operator and ...

Efficient Importance Sampling Algorithm Applied to the Performance Analysis of Wireless Communication Systems Estimation

When assessing the performance of wireless communication systems operati...

Consensus-based rare event estimation

In this paper, we introduce a new algorithm for rare event estimation ba...

Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach

We propose a novel alternative approach to our previous work (Ben Hammou...

The ensemble Kalman filter for rare event estimation

We present a novel sampling-based method for estimating probabilities of...

A Universal Splitting Estimator for the Performance Evaluation of Wireless Communications Systems

We propose a unified rare-event estimator for the performance evaluation...

Please sign up or login with your details

Forgot password? Click here to reset