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Importance sampling and stratification for copula models. (English) Zbl 1405.65005
Dick, Josef (ed.) et al., Contemporary computational mathematics – a celebration of the 80th birthday of Ian Sloan. In 2 volumes. Cham: Springer (ISBN 978-3-319-72455-3/hbk; 978-3-319-72456-0/ebook). 75-96 (2018).
Summary: An importance sampling approach for sampling from copula models is introduced. The proposed algorithm improves Monte Carlo estimators when the functional of interest depends mainly on the behaviour of the underlying random vector when at least one of its components is large. Such problems often arise from dependence models in finance and insurance. The importance sampling framework we propose is particularly easy to implement for Archimedean copulas. We also show how the proposal distribution of our algorithm can be optimized by making a connection with stratified sampling. In a case study inspired by a typical insurance application, we obtain variance reduction factors sometimes larger than 1000 in comparison to standard Monte Carlo estimators when both importance sampling and quasi-Monte Carlo methods are used.
For the entire collection see [Zbl 1398.65010].

65C05 Monte Carlo methods
62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text: DOI
[1] Cambou, M., Hofert, M., Lemieux, C.: Quasi-random numbers for copula models. Stat. Comput. 27(5), 1307-1329 (2017) · Zbl 06737713
[2] Chan, J., Kroese, D.: Efficient estimation of large portfolio loss probabilities in t-copula models. Eur. J. Oper. Res. 205(2), 361-367 (2010) · Zbl 1188.91231
[3] Choe, G., Jang, H.: Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas. Insurance: Math. Econ. 48(2), 205-213 (2011) · Zbl 1233.91296
[4] Derflinger, G., Hörmann, W., Leydold, J.: Random variate generation by numerical inversion when only the density is known. ACM Trans. Model. Comput. Simul. 20(4), 1-25 (2010) · Zbl 1386.65028
[5] Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010) · Zbl 1202.60001
[6] Ghalanos, A.: rugarch: Univariate GARCH models (2015). R package version 1.3-6
[7] Ghalanos, A.: spd: Semi-Parametric Distribution (2015). R package version 2.0-1
[8] Glasserman, P., Li, J.: Importance sampling for portfolio credit risk. Manag. Sci. 51(11), 1643-1656 (2005) · Zbl 1232.91621
[9] Hesterberg, T.: Weighted average importance sampling and defensive mixture distributions. Technometrics 37(2), 185-194 (1995) · Zbl 0822.62002
[10] Hofert, M., Mächler, M., et al.: Nested Archimedean copulas meet R: The nacopula package. J. Stat. Softw. 39(9), 1-20 (2011)
[11] Hofert, M., Kojadinovic, I., Maechler, M., Yan, J.: copula: Multivariate Dependence with Copulas (2016). R package version 0.999-15
[12] Huang, P., Subramanian, D., Xu, J.: An importance sampling method for portfolio CVaR estimation with Gaussian copula models. In: Proceedings of the 2010 Winter Simulation Conference (WSC), pp. 2790-2800 (2010)
[13] Lemieux, C.: Monte Carlo and Quasi-Monte-Carlo Sampling. Springer, New York (2009) · Zbl 1269.65001
[14] Marshall, A., Olkin, I.: Families of multivariate distributions. J. Am. Stat. Assoc. 83(403), 834-841 (1988) · Zbl 0683.62029
[15] McNeil, A.J., Frey, R.: Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. Empir. Financ. 7(3), 271-300 (2000)
[16] McNeil, A., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press, Princeton (2005) · Zbl 1089.91037
[17] Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006) · Zbl 1152.62030
[18] Ruckdeschel, P., Kohl, M., Stabla, T., Camphausen, F.: S4 classes for distributions. R News 6(2), 2-6 (2006)
[19] Sak, H., Hörmann, W., Leydold, J.: Efficient risk simulations for linear asset portfolios in the t-copula model. Eur. J. Oper. Res. 202(3), 802-809 (2010) · Zbl 1176.91150
[20] Sobol, I.: On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7(4), 86-112 (1967) · Zbl 0185.41103
[21] Tasche, D.: Capital allocation to business units and sub-portfolios: the Euler principle. In: Resti, A. (ed.) Pillar II in the New Basel Accord: The Challenge of Economic Capital, pp. 423-453. Risk Books, London (2008)
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