Sampling Archimedean copulas

The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed. Specific focus is put on large dimensions, where methods involving generator derivatives are not applicable. Additionally, new conditions under which Archimedean copulas can be mixed to construct nested Archimedean copulas are presented. Moreover, for some Archimedean families, direct sampling algorithms are given. For other families, sampling algorithms based on numerical inversion of Laplace transforms are suggested. For this purpose, the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm are compared in terms of precision and runtime. Examples are given, including both exchangeable and nested Archimedean copulas.

[1]  A W Gillies,et al.  Einfuhrung In Theorie Und Anwendung Der Laplace-Transformation , 1959 .

[2]  A. McNeil.  Sampling nested Archimedean copulas , 2008 .

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

[4]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

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

[6]  Ward Whitt,et al.  On the Laguerre Method for Numerically Inverting Laplace Transforms , 1996, INFORMS J. Comput..

[7]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[8]  G. Doetsch,et al.  Einführung in Theorie und Anwendung der Laplace-Transformation : ein Lehrbuch für Studierende der Mathematik, Physik und Ingenieurwissenschaft , 1958 .

[9]  D. P. Gaver,et al.  Observing Stochastic Processes, and Approximate Transform Inversion , 1966, Oper. Res..

[10]  J. Abate,et al.  Multi‐precision Laplace transform inversion , 2004 .

[11]  Niall Whelan,et al.  Sampling from Archimedean copulas , 2004 .

[12]  H. Joe.  Multivariate models and dependence concepts , 1998 .

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

[14]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

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

[16]  Peter P. Valko,et al.  Comparison of sequence accelerators forthe Gaver method of numerical Laplace transform inversion , 2004 .

[17]  C. Kimberling.  A probabilistic interpretation of complete monotonicity , 1974 .

[18]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[19]  P. Wynn,et al.  Sequence Transformations and their Applications. , 1982 .

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

[21]  StehfestHarald.  Remark on algorithm 368: Numerical inversion of Laplace transforms , 1970 .

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

[23]  F. Oberhettinger,et al.  Tables of Laplace Transforms , 1973 .

[24]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

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