A stochastic representation and sampling algorithm for nested Archimedean copulas

A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.

[1]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[2]  Marius Hofert,et al.  Nested Archimedean Copulas Meet R: The nacopula Package , 2011 .

[3]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

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

[5]  Luc Devroye,et al.  Random variate generation for exponentially and polynomially tilted stable distributions , 2009, TOMC.

[6]  Marius Hofert,et al.  Efficiently sampling nested Archimedean copulas , 2011, Comput. Stat. Data Anal..

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

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

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

[10]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

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

[12]  M. Hofert,et al.  CDO pricing with nested Archimedean copulas , 2011 .

[13]  N. H. Abel.  Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, daß f(z, f (x,y)) eine symmetrische Function von z, x und y ist. , 1826 .

[14]  Adrienne W. Kemp.  Efficient Generation of Logarithmically Distributed Pseudo‐Random Variables , 1981 .

[15]  Harry Joe,et al.  Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions , 1996 .

[16]  G. Christoph,et al.  Scaled Sibuya distribution and discrete self-decomposability , 2000 .

[17]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[18]  Marius Hofert,et al.  Sampling Exponentially Tilted Stable Distributions , 2011, TOMC.

[19]  Masaaki Sibuya,et al.  Generalized hypergeometric, digamma and trigamma distributions , 1979 .

[20]  Harry Joe,et al.  Modelling heavy-tailed count data using a generalised Poisson-inverse Gaussian family , 2009 .

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

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

[23]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[24]  Mark Trede,et al.  Goodness-of-fit tests for parametric families of Archimedean copulas , 2008 .

[25]  M. Deipenbrock.  RHEINISCH-WESTFÄLISCHE TECHNISCHE HOCHSCHULE AACHEN , 2012 .

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