Monte Carlo methods. Vol. 1: basics
Monte Carlo methods. Vol. 1: basics
Importance sampling for stochastic simulations
Management Science
The asymptotic efficiency of simulation estimators
Operations Research
A Unified Framework for Simulating Markovian Models of Highly Dependable Systems
IEEE Transactions on Computers
Importance sampling for the simulation of highly reliable Markovian systems
Management Science
Fast simulation of rare events in queueing and reliability models
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Simulating GI/GI/1 queues and insurance risk processes with subexponential distributions
Proceedings of the 32nd conference on Winter simulation
Multilevel Splitting for Estimating Rare Event Probabilities
Operations Research
On Numerical Problems in Simulations of Highly Reliable Markovian Systems
QEST '04 Proceedings of the The Quantitative Evaluation of Systems, First International Conference
On the efficiency of RESTART for multidimensional state systems
ACM Transactions on Modeling and Computer Simulation (TOMACS)
New measures of robustness in rare event simulation
WSC '05 Proceedings of the 37th conference on Winter simulation
Strongly efficient estimators for light-tailed sums
valuetools '06 Proceedings of the 1st international conference on Performance evaluation methodolgies and tools
Rare events, splitting, and quasi-Monte Carlo
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error
Queueing Systems: Theory and Applications
Approximate zero-variance simulation
Proceedings of the 40th Conference on Winter Simulation
Introduction to Rare Event Simulation
Introduction to Rare Event Simulation
Efficiency of high-order moment estimates
IEEE Transactions on Signal Processing
Approximate zero-variance simulation
Proceedings of the 40th Conference on Winter Simulation
Importance sampling simulations of phase-type queues
Winter Simulation Conference
Efficient calculation of rare event probabilities in Markovian queueing networks
Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
Optimal Monte Carlo integration with fixed relative precision
Journal of Complexity
Small Variance Estimators for Rare Event Probabilities
ACM Transactions on Modeling and Computer Simulation (TOMACS) - Special Issue on Monte Carlo Methods in Statistics
Probabilistic bounded relative error for rare event simulation learning techniques
Proceedings of the Winter Simulation Conference
Efficient importance sampling under partial information
Proceedings of the Winter Simulation Conference
Markov chain importance sampling with applications to rare event probability estimation
Statistics and Computing
Rare event simulation techniques
Proceedings of the Winter Simulation Conference
Graph reductions to speed up importance sampling-based static reliability estimation
Proceedings of the Winter Simulation Conference
Proceedings of the Winter Simulation Conference
Rare-event simulation for stochastic recurrence equations with heavy-tailed innovations
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Computers and Operations Research
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The asymptotic robustness of estimators as a function of a rarity parameter, in the context of rare-event simulation, is often qualified by properties such as bounded relative error (BRE) and logarithmic efficiency (LE), also called asymptotic optimality. However, these properties do not suffice to ensure that moments of order higher than one are well estimated. For example, they do not guarantee that the variance of the empirical variance remains under control as a function of the rarity parameter. We study generalizations of the BRE and LE properties that take care of this limitation. They are named bounded relative moment of order k (BRM-k) and logarithmic efficiency of order k (LE-k), where k ≥ 1 is an arbitrary real number. We also introduce and examine a stronger notion called vanishing relative centered moment of order k, and exhibit examples where it holds. These properties are of interest for various estimators, including the empirical mean and the empirical variance. We develop (sufficient) Lyapunov-type conditions for these properties in a setting where state-dependent importance sampling (IS) is used to estimate first-passage time probabilities. We show how these conditions can guide us in the design of good IS schemes, that enjoy convenient asymptotic robustness properties, in the context of random walks with light-tailed and heavy-tailed increments. As another illustration, we study the hierarchy between these robustness properties (and a few others) for a model of highly reliable Markovian system (HRMS) where the goal is to estimate the failure probability of the system. In this setting, for a popular class of IS schemes, we show that BRM-k and LE-k are equivalent and that these properties become strictly stronger when k increases. We also obtain a necessary and sufficient condition for BRM-k in terms of quantities that can be readily computed from the parameters of the model.