Efficient Simulation for Large Deviation Probabilities of Sums of Heavy-Tailed Increments

Blanchet,; Liu, Ming

doi:10.1109/wsc.2006.323156

Cited by 5 publications

(19 citation statements)

References 20 publications

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“…Approximating the h-transform closely is crucial for the sequential (state-dependent) importance sampling methods of Blanchet and Glynn [4] and Blanchet and Liu [6], [7] to be strongly efficient. This requires sharp and easily computable analytic approximations of α and p, provided by the Pakes-Veraberbeke theorem [1, p. 296] in [4] and provided by Rozovskiǐ's theorem [18] in [7].…”

Section: Other Methods Related Work and Discussionmentioning

confidence: 99%

“…We next show that we can bypass the normalizing constant by using SISR, which also enables us to weaken and generalize (6) to As noted in Section 1, the normalizing constant (i.e.…”

Section: Implementing a Target Importance Measure By Sisrmentioning

confidence: 92%

“…In [7], a computationally expensive discretization scheme, with partition width 1/n, is used to implement the state-dependent importance sampling scheme based on the asymptotic approximation (6) in the case of regularly varying random walks. In [7], a computationally expensive discretization scheme, with partition width 1/n, is used to implement the state-dependent importance sampling scheme based on the asymptotic approximation (6) in the case of regularly varying random walks.…”

Section: Implementing a Target Importance Measure By Sisrmentioning

confidence: 99%

“…To achieve strong efficiency, Blanchet and Glynn [4] and Blanchet and Liu [6] made use of approximations of Doob's h-transform to develop an importance sampling method for computing P(A) when the event A is related to a Markov chain Y k that has transition probability densities p k (· | Y k−1 ) with respect to some measure ν. Letting h k (Y k …”

Section: Introductionmentioning

confidence: 99%

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Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012

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We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

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Section: Other Methods Related Work and Discussionmentioning

confidence: 99%

“…We next show that we can bypass the normalizing constant by using SISR, which also enables us to weaken and generalize (6) to As noted in Section 1, the normalizing constant (i.e.…”

Section: Implementing a Target Importance Measure By Sisrmentioning

confidence: 92%

Section: Implementing a Target Importance Measure By Sisrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012

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“…Dupuis et al [8] also propose algorithms that have bounded relative error in estimating these probabilities, however their proposed algorithms are state-dependent. Blanchet and Liu [7] consider the problem P (S n > nu) as n → ∞ and u > 0 is fixed. For this and related problems they develop a novel state-dependent change of measure that has the bounded relative error property.…”

Section: Introductionmentioning

confidence: 99%

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Juneja

2007

Queueing Syst

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Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter 'residual' probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.Keywords Bounded relative error · Asymptotically zero relative error · Complexity · Rare event simulation · Regular variation · M/G/1 queue · Subexponential

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State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables

Amar

Rached²,

Haji-Ali³

et al. 2023

Stat Comput

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Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.

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Efficient Simulation for Large Deviation Probabilities of Sums of Heavy-Tailed Increments

Cited by 5 publications

References 20 publications

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables

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