2 research outputs found
A Fast Simulation Method for the Sum of Subexponential Distributions
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
An Improved Hazard Rate Twisting Approach for the Statistic of the Sum of Subexponential Variates (Extended Version)
In this letter, we present an improved hazard rate twisting technique for the
estimation of the probability that a sum of independent but not necessarily
identically distributed subexponential Random Variables (RVs) exceeds a given
threshold. Instead of twisting all the components in the summation, we propose
to twist only the RVs which have the biggest impact on the right-tail of the
sum distribution and keep the other RVs unchanged. A minmax approach is
performed to determine the optimal twisting parameter which leads to an
asymptotic optimality criterion. Moreover, we show through some selected
simulation results that our proposed approach results in a variance reduction
compared to the technique where all the components are twisted