Copulas with Prescribed Correlation Matrix

Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

Consider the convex set R n of semi positive definite matrices of order n with diagonal \((1,\ldots,1).\) If μ is a distribution in \(\mathbb{R}^{n}\) with second moments, denote by R(μ) ∈ R n its correlation matrix. Denote by C n the set of distributions in [0, 1] n with all margins uniform on [0, 1] (called copulas). The paper proves that \(\mu \mapsto R(\mu )\) is a surjection from C n on R n if n ≤ 9. It also studies the Gaussian copulas μ such that R(μ) = R for a given R ∈ R n . 

Keywords

Correlation Matrix Extreme Point Orthogonal Projection Symmetric Matrice Stochastic Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Devroye, G. Letac, Copulas in three dimensions with prescribed correlations (2010) [arXiv 1004.3146]Google Scholar
  2. 2.
    M. Falk, A simple approach to the generation of uniformly distributed random variables with prescribed correlations. Commun. Stat. Simul. Comp. 28, 785–791 (1999)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Gasper, Positivity and the convolution structure for Jacobi series. Ann. Math. 93, 112–118 (1971)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    G. Gasper, Banach algebra for Jacobi series and positivity of a kernel. Ann. Math. 95, 261–280 (1972)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    A.E. Koudou, Problèmes de marges et familles exponentielles naturelles. Thèse, Université Paul Sabatier, Toulouse, 1995Google Scholar
  6. 6.
    A.E. Koudou, Probabilités de Lancaster. Expositiones Math. 14, 247–275 (1996)MATHMathSciNetGoogle Scholar
  7. 7.
    G. Letac, Lancaster probabilities and Gibbs sampling. Stat. Sci. 23, 187–191 (2008)CrossRefGoogle Scholar
  8. 8.
    B. Ycart, Extreme points in convex sets of symmetric matrices. Proc. Am. Math. Soc. 95(4), 607–612 (1985)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Equipe de Statistique et ProbabilitésUniversité de ToulouseToulouseFrance

Personalised recommendations