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Large deviations for heavy-tailed factor models
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-6608-0715
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.

Place, publisher, year, edition, pages
2009. Vol. 79, no 3, 304-311 p.
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-7579DOI: 10.1016/j.spl.2008.08.011ISI: 000263424000005Scopus ID: 2-s2.0-58149502560OAI: oai:DiVA.org:kth-7579DiVA: diva2:12649
Note
QC 20100811Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Some asymptotic results in dependence modelling
Open this publication in new window or tab >>Some asymptotic results in dependence modelling
2007 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two papers, both devoted to the study of asymptotics in dependence modelling.

The first paper studies large deviation probabilities for a sum of dependent random variables, where 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 behaviour are found and the resulting approximation is evaluated in a simple special case. The results are then applied to stochastic processes with the same structure. Based on the results of the first part of the paper, it is concluded that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process.

The second paper studies the asymptotic eigenvalue distribution of the exponentially weighted moving average (EWMA) covariance estimator. Equations for the limiting eigenvalue density and the boundaries of its support are found using the Marchenko-Pastur theorem.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. iii p.
Series
Trita-MAT, ISSN 1401-2286 ; 2007:01
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-4519 (URN)978-91-7178-792-7 (ISBN)
Presentation
2007-11-05, 3733, Matematiska Institutionen, Lindstedtsvägen 25, Stockholm, 15:30
Opponent
Supervisors
Note
QC 20101119Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2010-11-19Bibliographically approved
2. 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.
Series
Trita-MAT. MS, 09:11
National Category
Probability Theory and Statistics
Identifiers
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)
Opponent
Supervisors
Note
QC 20100811Available from: 2009-10-14 Created: 2009-10-13 Last updated: 2010-08-11Bibliographically approved

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