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On Importance Sampling with Mixtures for Random Walks with Heavy Tails
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0001-9210-121X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2012 (English)In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, Vol. 22, no 2, 8- p.Article in journal (Refereed) Published
Abstract [en]

State-dependent importance sampling algorithms based on mixtures are considered. The algorithms are designed to compute tail probabilities of a heavy-tailed random walk. The increments of the random walk are assumed to have a regularly varying distribution. Sufficient conditions for obtaining bounded relative error are presented for rather general mixture algorithms. Two new examples, called the generalized Pareto mixture and the scaling mixture, are introduced. Both examples have good asymptotic properties and, in contrast to some of the existing algorithms, they are very easy to implement. Their performance is illustrated by numerical experiments. Finally, it is proved that mixture algorithms of this kind can be designed to have vanishing relative error.

Place, publisher, year, edition, pages
2012. Vol. 22, no 2, 8- p.
Keyword [en]
Rare event simulation, heavy tails, importance sampling
National Category
Probability Theory and Statistics
URN: urn:nbn:se:kth:diva-11269DOI: 10.1145/2133390.2133392ISI: 000302131400002ScopusID: 2-s2.0-84859453675OAI: diva2:271897
Swedish Research Council, 621-2008-4944
QC 20100811Available from: 2009-10-13 Created: 2009-10-13 Last updated: 2012-05-03Bibliographically approved
In thesis
1. On Importance Sampling and Dependence Modeling
Open this publication in new window or tab >>On Importance Sampling and Dependence Modeling
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers.

In the first paper, Monte Carlo simulation for tail probabilities of heavy-tailed random walks is considered. Importance sampling algorithms are constructed by using mixtures of the original distribution with some other state-dependent distributions. Sufficient conditions under which the relative error of such algorithms is bounded are found, and the bound is calculated. A new mixture algorithm based on scaling of the original distribution is presented and compared to existing algorithms.

In the second paper, Monte Carlo simulation of quantiles is treated. It is shown that by using importance sampling algorithms developed for tail probability estimation, efficient quantile estimators can be obtained. A functional limit of the quantile process under the importance sampling measure is found, and the variance of the limit process is calculated for regularly varying distributions. The procedure is also applied to the calculation of expected shortfall. The algorithms are illustrated numerically for a heavy-tailed random walk.

In the third paper, large deviation probabilities for a sum of dependent random variables are derived. The dependence stems from a few underlying random variables, so-called factors. Each summand is composed of two parts: an idiosyncratic part and a part given by the factors. Conditions under which both factors and idiosyncratic components contribute to the large deviation behavior are found, and the resulting approximation is evaluated in a simple example.

In the fourth paper, the asymptotic eigenvalue distribution of the exponentially weighted moving average covariance estimator is studied. Equations for the asymptotic spectral density and the boundaries of its support are found using the Marchenko-Pastur theorem.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vi, 13 p.
Trita-MAT. MS, 09:11
National Category
Probability Theory and Statistics
urn:nbn:se:kth:diva-11272 (URN)978-91-7415-433-7 (ISBN)
Public defence
2009-10-23, D2, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
QC 20100811Available from: 2009-10-14 Created: 2009-10-13 Last updated: 2010-08-11Bibliographically approved

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