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On Importance Sampling and Dependence Modeling
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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.
Series
Trita-MAT. MS, 09:11
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-11272ISBN: 978-91-7415-433-7 (print)OAI: oai:DiVA.org:kth-11272DiVA: diva2:271904
Public defence
2009-10-23, D2, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100811Available from: 2009-10-14 Created: 2009-10-13 Last updated: 2010-08-11Bibliographically approved
List of papers
1. On Importance Sampling with Mixtures for Random Walks with Heavy Tails
Open this publication in new window or tab >>On Importance Sampling with Mixtures for Random Walks with Heavy Tails
2012 (English)In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, 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.

Keyword
Rare event simulation, heavy tails, importance sampling
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-11269 (URN)10.1145/2133390.2133392 (DOI)000302131400002 ()2-s2.0-84859453675 (Scopus ID)
Funder
Swedish Research Council, 621-2008-4944
Note
QC 20100811Available from: 2009-10-13 Created: 2009-10-13 Last updated: 2017-12-12Bibliographically approved
2. Efficient calculation of risk measures by importance sampling -- the heavy tailed case
Open this publication in new window or tab >>Efficient calculation of risk measures by importance sampling -- the heavy tailed case
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Computation of extreme quantiles and tail-based risk measures using standard Monte Carlo simulation can be inefficient. A method to speed up computations is provided by importance sampling. We show that importance sampling algorithms, designed for efficient tail probability estimation, can significantly improve Monte Carlo estimators of tail-based risk measures. In the heavy-tailed setting, when the random variable of interest has a regularly varying distribution, we provide sufficient conditions for the asymptotic relative error of importance sampling estimators of risk measures, such as Value-at-Risk and expected shortfall, to be small. The results are illustrated by some numerical examples.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-11271 (URN)
Note
QC 20100811Available from: 2009-10-13 Created: 2009-10-13 Last updated: 2010-08-11Bibliographically approved
3. Large deviations for heavy-tailed factor models
Open this publication in new window or tab >>Large deviations for heavy-tailed factor models
2009 (English)In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 79, no 3, 304-311 p.Article in journal (Refereed) Published
Abstract [en]

We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. Depending on the regions considered, probabilities are determined by different parts of the model.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-7579 (URN)10.1016/j.spl.2008.08.011 (DOI)000263424000005 ()2-s2.0-58149502560 (Scopus ID)
Note
QC 20100811Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2017-12-14Bibliographically approved
4. The asymptotic spectrum of the EWMA covariance estimator
Open this publication in new window or tab >>The asymptotic spectrum of the EWMA covariance estimator
2007 (English)In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, no 385, 621-630 p.Article in journal (Refereed) Published
Abstract [en]

The exponentially weighted moving average (EWMA) covariance estimator is a standard estimator for financial time series, and its spectrum can be used for so-called random matrix filtering. Random matrix filtering using the spectrum of the sample covariance matrix is an established tool in finance and signal detection and the EWMA spectrum can be used analogously. In this paper, the asymptotic spectrum of the EWMA covariance estimator is calculated using the Mar enko-Pastur theorem. Equations for the spectrum and the boundaries of the support of the spectrum are obtained and solved numerically. The spectrum is compared with covariance estimates using simulated i.i.d. data and log-returns from a subset of stocks from the S&P 500. The behaviour of the EWMA estimator in this limited empirical study is similar to the results in previous studies of sample covariance matrices. Correlations in the data are found to only affect a small part of the EWMA spectrum, suggesting that a large part may be filtered out

Keyword
EWMA; random matrix theory; covariance estimation; noise
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-7578 (URN)10.1016/j.physa.2007.07.030 (DOI)000250491900021 ()2-s2.0-34548759639 (Scopus ID)
Note
QC 20100811Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2017-12-14Bibliographically approved

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