Quasi-random numbers for copula models

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.

[1]  Alexander J. McNeil,et al.  Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions , 2009, 0908.3750.

[2]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[3]  Alexander J. McNeil,et al.  Estimators for Archimedean copulas in high dimensions , 2012, 1207.1708.

[4]  Christiane Lemieux,et al.  Generalized Halton sequences in 2008: A comparative study , 2009, TOMC.

[5]  Edmund Hlawka,et al.  Über eine Transformation von gleichverteilten Folgen II , 2005, Computing.

[6]  F. Pillichshammer,et al.  Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , 2010 .

[7]  Emiliano A. Valdez,et al.  Simulating from Exchangeable Archimedean Copulas , 2007, Commun. Stat. Simul. Comput..

[8]  Roger M. Cooke,et al.  Sampling algorithms for generating joint uniform distributions using the vine-copula method , 2007, Comput. Stat. Data Anal..

[9]  J. L. Nolan.  Stable Distributions. Models for Heavy Tailed Data , 2001 .

[10]  J. M. Sek,et al.  On the L2-discrepancy for anchored boxes , 1998 .

[11]  P. Embrechts,et al.  Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management , 2003 .

[12]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[13]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[14]  P. Embrechts,et al.  Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls , 2002 .

[15]  I. Olkin,et al.  Families of Multivariate Distributions , 1988 .

[16]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[17]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[18]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[19]  Ronald Cools,et al.  Using Box-Muller with Low Discrepancy Points , 2006, ICCSA.

[20]  A. Owen.  Monte Carlo Variance of Scrambled Net Quadrature , 1997 .

[21]  Jürgen Hartinger,et al.  Quasi-Monte Carlo algorithms for unbounded, weighted integration problems , 2004, J. Complex..

[22]  C. Lemieux.  Monte Carlo and Quasi-Monte Carlo Sampling , 2009 .

[23]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[24]  Jirí Matousek,et al.  On the L2-Discrepancy for Anchored Boxes , 1998, J. Complex..

[25]  C. Aistleitner,et al.  Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality , 2014, 1406.0230.

[26]  H. Joe.  Dependence Modeling with Copulas , 2014 .

[27]  Fred J. Hickernell,et al.  Algorithm 823: Implementing scrambled digital sequences , 2003, TOMS.

[28]  A. Owen.  Multidimensional variation for quasi-Monte Carlo , 2004 .

[29]  Marius Hofert,et al.  Construction and Sampling of Nested Archimedean Copulas , 2010 .

[30]  Dirk Tasche,et al.  Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle , 2007, 0708.2542.

[31]  Alexander J. McNeil,et al.  Likelihood inference for Archimedean copulas in high dimensions under known margins , 2012, J. Multivar. Anal..

[32]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[33]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[34]  A. Owen.  Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences , 1995 .

[35]  Fabrizio Durante,et al.  Copula Theory and Its Applications , 2010 .

[36]  William J. Morokoff.  Generating Quasi-Random Paths for Stochastic Processes , 1998, SIAM Rev..

[37]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[38]  H. Bock,et al.  Copulas and stochastic processes , 2003 .

[39]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[40]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[41]  Art B. Owen,et al.  Variance with alternative scramblings of digital nets , 2003, TOMC.

[42]  Art B. Owen,et al.  Variance and discrepancy with alternative scramblings , 2002 .

[43]  J Lin.  FUNCTIONS OF BOUNDED VARIATION OF ORDER 3 , 1957 .

[44]  A. Owen.  Scrambled net variance for integrals of smooth functions , 1997 .

[45]  Florence Wu.  Simulating Exchangeable Multivariate Archimedean Copulas and its Applications ∗ , 2005 .

[46]  Edmund Hlawka.  Über die Diskrepanz mehrdimensionaler Folgen mod. 1 , 1961 .

[47]  F. Pillichshammer,et al.  Digital Nets and Sequences: Nets and sequences , 2010 .

[48]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[49]  Paul Embrechts,et al.  Quantitative Risk Management , 2011, International Encyclopedia of Statistical Science.

[50]  M. Hofert.  Sampling Nested Archimedean Copulas: with Applications to CDO Pricing , 2010 .

[51]  A. McNeil,et al.  Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications , 2012 .

[52]  Colas Schretter,et al.  Monte Carlo and Quasi-Monte Carlo Methods , 2016 .

[53]  C. Baxa.  Calculation of improper integrals using uniformly distributed sequences , 2005 .

[54]  G. Constantine,et al.  A Multivariate Faa di Bruno Formula with Applications , 1996 .

[55]  R. Nelsen.  An Introduction to Copulas , 1998 .