×

zbMATH — the first resource for mathematics

Quasi-random numbers for copula models. (English) Zbl 06737713
Summary: The present work addresses the question how sampling algorithms for commonly applied copula models can be adapted to account for quasi-random numbers. Besides sampling methods such as the conditional distribution method (based on a one-to-one transformation), it is also shown that typically faster sampling methods (based on stochastic representations) can be used to improve upon classical Monte Carlo methods when pseudo-random number generators are replaced by quasi-random number generators. This opens the door to quasi-random numbers for models well beyond independent margins or the multivariate normal distribution. Detailed examples (in the context of finance and insurance), illustrations and simulations are given and software has been developed and provided in the R packages copula and qrng.

MSC:
62H99 Multivariate analysis
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aistleitner, C; Dick, J, Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality, Acta Arith., 167, 143-171, (2015) · Zbl 1326.11038
[2] Caflisch, R, Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7, 1-49, (1998) · Zbl 0949.65003
[3] Cambanis, S; Huang, S; Simons, G, On the theory of elliptically contoured distributions, J. Multivar. Anal., 11, 368-385, (1981) · Zbl 0469.60019
[4] Constantine, G; Savits, T, A multivariate faa di bruno formula with applications, Trans. Am. Math. Soc., 348, 503-520, (1996) · Zbl 0846.05003
[5] Cranley, R; Patterson, TNL, Randomization of number theoretic methods for multiple integration, SIAM J. Numer. Anal., 13, 904-914, (1976) · Zbl 0354.65016
[6] Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010) · Zbl 1282.65012
[7] Embrechts, P; McNeil, AJ; Straumann, D, Correlation and dependence in risk management: properties and pitfalls, (2002), Cambridge
[8] Embrechts, P; Lindskog, F; McNeil, AJ; Rachev, S (ed.), Modelling dependence with copulas and applications to risk management, 329-384, (2003), Boston
[9] Fang, K.T., Kotz, S., Ng, K.W.: Symmetric Multivariate and Related Distributions. Chapman & Hall, Boca Raton (1990) · Zbl 0699.62048
[10] Faure, H, Discrépance des suites associées à un système de numération (en dimension \(s\)), Acta Arith., 41, 337-351, (1982) · Zbl 0442.10035
[11] Faure, H; Lemieux, C, Generalized halton sequence in 2008: a comparative study, ACM Trans. Model. Comput. Simul., 19, 15, (2009) · Zbl 1288.65003
[12] Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2004) · Zbl 1038.91045
[13] Halton, JH, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math., 2, 84-90, (1960) · Zbl 0090.34505
[14] Hartinger, J; Kainhofer, R; Tichy, R, Quasi-monce Carlo algorithms for unbounded, weighted integration problems, J. Complex., 20, 654-668, (2004) · Zbl 1072.65001
[15] Hlawka, E, Über die diskrepanz mehrdimensionaler folgen mod 1, Math. Z., 77, 273-284, (1961) · Zbl 0112.27803
[16] Hlawka, E; Mück, R, Über eine transformation von gleichverteilten folgen II, Computing, 9, 127-138, (1972) · Zbl 0245.10039
[17] Hofert, M.: Sampling nested Archimedean copulas with applications to CDO pricing. PhD thesis, University of Ulm (2010) · Zbl 1232.91690
[18] Hofert, M; Mächler, M; McNeil, AJ, Likelihood inference for Archimedean copulas in high dimensions under known margins, J. Multivar. Anal., 110, 133-150, (2012) · Zbl 1244.62073
[19] Hofert, M; Mächler, M; McNeil, AJ, Archimedean copulas in high dimensions: estimators and numerical challenges motivated by financial applications, Journal de la Société Française de Statistique, 154, 25-63, (2013) · Zbl 1316.62070
[20] Hong, HS; Hickernell, FJ, Algorithm 823: implementing scrambled digital sequences, ACM Trans. Math. Softw., 29, 95-109, (2003) · Zbl 1068.11049
[21] Jaworski, P., Durante, F., Härdle, W.K., Rychlik, T.: Copula Theory and Its Applications. Lecture Notes in Statistics—Proceedings. Springer, Heidelberg (2010) · Zbl 1194.62077
[22] Joe, H.: Dependence Modeling with Copulas. Chapman & Hall, Boca Raton (2014) · Zbl 1346.62001
[23] Kurowicka, D; Cooke, RM, Sampling algorithms for generating joint uniform distributions using the vine-copula method, Comput. Stat. Data Anal., 51, 2889-2906, (2007) · Zbl 1161.62363
[24] Lemieux, C.: Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics. Springer, New York (2009) · Zbl 1269.65001
[25] Marshall, AW; Olkin, I, Families of multivariate distributions, J. Am. Stat. Assoc., 83, 834-841, (1988) · Zbl 0683.62029
[26] Matousěk, J, On the \({L_2}\)-discrepancy for anchored boxes, J. Complex., 14, 527-556, (1998) · Zbl 0942.65021
[27] McNeil, A., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press, Princeton (2005) · Zbl 1089.91037
[28] McNeil, AJ; Nešlehová, J, Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_{1}\)-norm symmetric distributions, Ann. Stat., 37, 3059-3097, (2009) · Zbl 1173.62044
[29] Morokoff, W, Generating quasi-random paths for stochastic processes, SIAM Rev., 40, 765-788, (1998) · Zbl 0913.65143
[30] Morokoff, W; Caflisch, R, Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput., 15, 1251-1279, (1994) · Zbl 0815.65002
[31] Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006) · Zbl 1152.62030
[32] Niederreiter, H, Point sets and sequences with small discrepancy, Monatshefte für Mathematik, 104, 273-337, (1987) · Zbl 0626.10045
[33] Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia (1992) · Zbl 0761.65002
[34] Nolan, J.: Stable distributions—models for heavy tailed data. http://academic2.american.edu/jpnolan/stable/chap1 (2014) · Zbl 0469.60019
[35] Owen, AB; Niederreiter, H (ed.); Shiue, PJS (ed.), Randomly permuted \((t, m, s)\)-nets and \((t, s)\)-sequences, No. 106, 299-317, (1995), New York · Zbl 0831.65024
[36] Owen, AB, Monte Carlo variance of scrambled equidistribution quadrature, SIAM J. Numer. Anal., 34, 1884-1910, (1997) · Zbl 0890.65023
[37] Owen, AB, Scrambled net variance for integrals of smooth functions, Ann. Stat., 25, 1541-1562, (1997) · Zbl 0886.65018
[38] Owen, AB, Variance and discrepancy with alternative scramblings, ACM Trans. Model. Comput. Simul., 13, 363-378, (2003) · Zbl 1390.65037
[39] Owen, A.B.: Multidimensional variation for quasi-Monte Carlo. In: Fan, J., Li, G. (eds.) International Conference on Statistics in honour of Professor Kai-Tai Fang’s 65th birthday, pp. 49-74. World Scientific Publications, Hackensack, NJ (2005)
[40] Pillards, T., Cools, R.: Using box-muller with low discrepancy points. In: ICCSA 2006, Lecture Notes in Computer Science, vol. 3984, pp 780-788. Springer, Berlin (2006) · Zbl 1175.65006
[41] Schmitz, V.: Copulas and stochastic processes. PhD thesis, Institute of Statistics, Aachen University (2003) · Zbl 0846.05003
[42] Sobol’, IM, On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comput. Math. Math. Phys., 7, 86-112, (1967) · Zbl 0185.41103
[43] Sobol’, IM, Calculation of improper integrals using uniformly distributed sequences, Sov. Math. Dokl., 14, 734-738, (1973) · Zbl 0283.41016
[44] Tasche, D; Resti, A (ed.), Capital allocation to business units and sub-portfolios: the Euler principle, 423-453, (2008), London
[45] Wu, F; Valdez, EA; Sheris, M, Simulating exchangeable multivariate Archimedean copulas and its applications, Commun. Stat., 36, 1019-1034, (2006) · Zbl 1126.62041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.