The transform likelihood ratio method for rare event simulation with heavy tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation ( change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method

The Restart/LRE Method for Rare Event Simulation

Using the LRE-algorithm for the evaluation of simulated data yields the stationary distribution function of an investigated random sequence and additionally the so-called local correlation coefficient, which represents relevant correlation evidence to be included in the error measure for controlling the simulation run time. In this paper a simplified LRE-algorithm is used to evaluate discrete sequences like the occupancy of finite buffer queueing systems G/G/1/N . It is shown how this algorithm is combined with the RESTART -method for an efficient rare event simulation. A multi-stage RESTART/LRE-algorithmhas been implemented as part of a stochastic simulation system and its performance has been verified by extensive simulations of the reference system M/M/1/N , whose properties including the local correlation coefficient can be described by analytical formulas. Approximate formulas for the optimal number of stages and the number of trials are given. The new algorithm has been successfu..