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[24] F Le Gland and N Oudjane A sequential algorithm that keeps the particle system alive In Henk Blom and John Lygeros, eds, Stochastic Hybrid Systems: Theory and Safety Critical Applications, Lecture Notes in Control and Information Sciences 337, pp 351 389 Springer, Berlin, 2006 [25] P L Ecuyer, V Demers, and B Tuf n Splitting for rare-event simulation In Proceedings of the 2006 Winter Simulation Conference, Monterey 2006, pp 137 148, December 2006 [26] P L Ecuyer, V Demers, and B Tuf n Rare events, splitting, and quasi-Monte Carlo ACM Transactions on Modeling and Computer Simulation, 17(2), 2007 Article 9 [27] P L Ecuyer and B Tuf n Splitting and weight windows to control the likelihood ratio in importance sampling In Proceedings of the 1st International Conference on Performance Evaluation Methodologies and Tools (ValueTools), Pisa 2006 , volume 180 of ACM International Conference Proceeding Series ACM, October 2006 Article 21 [28] V B Melas On the ef ciency of the splitting and roulette approach for sensitivity analysis In Proceedings of the 1997 Winter Simulation Conference, Atlanta 1997 , pp 269 274, December 1997 [29] H Soodak Pile kinetics In C Goodman, ed, The Science and Engineering of Nuclear Power , Volume 2, pp 89 102 Addison-Wesley, Reading, MA, 1949 [30] J Vill n-Altamirano Rare event RESTART simulation of two-stage networks Euroe pean Journal of Operations Research, 179(1): 148 159, 2007 [31] M Vill n-Altamirano and J Vill n-Altamirano RESTART: A straightforward e e method for fast simulation of rare events In Proceedings of the 1994 Winter Simulation Conference, Orlando 1994 , pp 282 289, December 1994 [32] M Vill n-Altamirano and J Vill n-Altamirano Analysis of RESTART simulation: e e Theoretical basis and sensitivity study European Transactions on Telecommunications, 13(4): 373 385, 2002 [33] M Vill n-Altamirano and J Vill n-Altamirano On the ef ciency of RESTART e e for multidimensional state systems ACM Transactions on Modeling and Computer Simulation, 16(3): 251 279, 2006
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Robustness properties and con dence interval reliability issues
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Peter W Glynn, Gerardo Rubino and Bruno Tuf n
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Introduction
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In this chapter, we discuss the robustness and reliability of the estimators of the probability of a rare event (or, more generally, of the expectation of some function of rare events) with respect to rarity: is the estimator accurate as rarity increases (recall that accuracy, when estimating small probabilities, focuses on relative rather than absolute errors) And what about the reliability (ie, the coverage) of the associated con dence interval If we parameterize the model with a (small) real such that the probability of the rare event considered decreases to zero as 0, we need to control the quality of the estimator as rarity increases, with respect to accuracy and coverage An estimator will be said to be robust (in different senses de ned hereafter) if its quality (ie, the gap with respect to the true value) is not signi cantly affected when 0 Similarly, an estimator is always accompanied with a con dence interval A reliable estimator is then an estimator for which the con dence interval coverage does not deteriorate as 0 Those two notions are different: one focuses on the error itself, the other on quality of the error estimation
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Rare Event Simulation using Monte Carlo Methods Edited by Gerardo Rubino and Bruno Tuffin 2009 John Wiley & Sons, Ltd ISBN: 978-0-470-77269-0
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ROBUSTNESS AND CONFIDENCE INTERVAL RELIABILITY
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To better illustrate this, let us start with the standard or crude estimator of the probability of a rare event Let be this probability and (Xi )1 i n be independently and identically distributed random variables such that Xi = 1 if the rare event occurs at the ith trial and 0 otherwise The standard estimator of STD is n = n 1 n Xi The sum n Xi is a binomial random variable with i=1 i=1 STD variance n (1 ), and the resulting con dence interval for , centered at n , at con dence level 1 , is (1 ) STD (1 ) STD n z1 /2 , n + z1 /2 n n where z1 /2 = 1 (1 /2) and is the standard normal cumulative distribution function The relative half-width RE of the con dence interval is therefore z1 /2 1 / n For a xed sample size n, this means that, as 0, the relative error of the estimation goes to in nity Therefore, the accuracy of the estimator deteriorates as 0 The absolute error given by the con dence inter val half-width z /2 (1 )/ n tends to 0 with , but at the much smaller rate than , so it does not give a good idea of the order of magnitude of the probability of interest In other words, in order to get a xed relative half-width RE = of the con dence interval as 0, one would have to increase the sample size (which usually means the simulation computating time) as n = (z1 /2 )2 1 , 2
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that is, in inverse proportion to The aim of rare event simulation is to construct estimators for which the relative error is kept under control as the event probability decreases to zero Such estimators are said to be robust, and families of robusness properties will be discussed in this chapter But looking only at the (theoretical) relative error, or some of its closely related notions introduced below, may be hazardous, or may only provide partial views of the possible problems When evaluating using some unbiased estimator n = n 1 n Xi , where the Xi are independently and identically (generally) i=1 distributed random variables with mean and variance 2 , not only is E( n ) = 2 unknown in practice, but so is its variance Var( n ) = n = 2 /n Generally 2 2: is estimated by the unbiased n
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This estimator is at least as sensitive to rarity as n itself Returning to the crude estimation of a probability by the average of STD Bernoulli random variables n , if n is much smaller than 1/ , the rare event will most likely not be observed (on average, an occurrence appears after 1/ 2 replications), leading to a con dence interval (0, 0) because n = n = 0 With the (very unlikely) assumption that we end up with exactly one occurrence
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