Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T11:39:10.812Z Has data issue: false hasContentIssue false
  • English
  • Français

On a construction of multivariate distributions given some multidimensional marginals

Part of: Distribution theory - Probability Multivariate analysis

Published online by Cambridge University Press:  07 August 2019

Nabil Kazi-Tani  and
Didier Rullière
Nabil Kazi-Tani*
Affiliation:
Université Lyon 1
Didier Rullière*
Affiliation:
Université Lyon 1
*
*Postal address: Laboratoire SAF, ISFA, Université Lyon 1, 50 Avenue Tony Garnier, F-69366 Lyon Cedex 07, France.
*Postal address: Laboratoire SAF, ISFA, Université Lyon 1, 50 Avenue Tony Garnier, F-69366 Lyon Cedex 07, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data does not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

Keywords

Multidimensional marginalcopulaelliptical distribution

MSC classification

Primary: 60E05: Distributions 60E10: Characteristic functions; other transforms
Secondary: 62H05: Characterization and structure theory 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 487 - 513
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom . 44, 182–198.CrossRefGoogle Scholar
Acar, E. F., Genest, C. and Nešlehová, J. (2012). Beyond simplified pair-copula constructions. J. Multivariate Anal. 110, 74–90.CrossRefGoogle Scholar
Baxter, J. R. and Chacon, R. V. (1974). Potentials of stopped distributions. Illinois J. Math . 18, 649–656.CrossRefGoogle Scholar
Brechmann, E. C. (2014). Hierarchical kendall copulas: Properties and inference. Canad. J. Statist. 42, 78–108.CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I. and Galichon, A. (2010). Quantile and probability curves without crossing. Econometrica 78, 1093–1125.Google Scholar
Cohen, L. (1984). Probability distributions with given multivariate marginals. J. Math. Phys. 25, 2402–2403.CrossRefGoogle Scholar
Cuadras, C. M. (1992). Probability distributions with given multivariate marginals and given dependence structure. J. Multivariate Anal. 42, 51–66.CrossRefGoogle Scholar
Dall’aglio, G., Kotz, S. and Salinetti, G. (eds) (1991). Advances in Probability Distributions with Given Marginals: Beyond the Copulas (Mathematics and Its Applications 67), Springer, Netherlands.CrossRefGoogle Scholar
Di Bernardino, E. and Rullière, D. (2016). On an asymmetric extension of multivariate archimedean copulas based on quadratic form. Depend. Model. 4, 328–347.Google Scholar
Dolati, A. and Úbeda-Flores, M. (2005). A method for constructing multivariate distributions with given bivariate margins. Braz. J. Prob. Statist. 19, 85–92.Google Scholar
Erdely, A. and González-Barrios, J. M. (2008). Exact distribution under independence of the diagonal section of the empirical copula. Kybernetika (Prague) 44, 826–845.Google Scholar
Fang, K., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions (Monogr. Statist. Appl. Prob. 36). Chapman and Hall, London.CrossRefGoogle Scholar
Frahm, G. (2004). Generalized elliptical distributions: theory and applications. Doctoral Thesis. Universität zu Köln.Google Scholar
Frahm, G., Junker, M. and Szimayer, A. (2003). Elliptical copulas: applicability and limitations. Statist. Prob. Lett. 63, 275–286.CrossRefGoogle Scholar
Genest, C., Queseda Molina, J. J. and Rodriguez Lallena, J. A. (1995). De l’impossibilité de construire des lois à marges multidimensionnelles données à partir de copules. C. R. Acad. Sci. Ser. I Math. 320, 723–726.Google Scholar
Gutmann, S., Kemperman, J. H. B., Reeds, J. A. and Shepp, L. A. (1991). Existence of probability measures with given marginals. Ann. Prob. 19, 1781–1797.CrossRefGoogle Scholar
Hofert, M. and Pham, D. (2013). Densities of nested archimedean copulas. J. Multivariate Anal. 118, 37–52.CrossRefGoogle Scholar
Joe, H. (1993). Parametric families of multivariate distributions with given margins. J. Multivariate Anal. 46, 262–282.CrossRefGoogle Scholar
Joe, H. (1994). Multivariate extreme-value distributions with applications to environmental data. Canad. J. Statist. 22, 47–64.CrossRefGoogle Scholar
Joe, H. (1996). Families of m-variate distributions with given margins and m(m–1)/2 bivariate dependence parameters (IMS Lecture Notes Monogr. Ser. 28), Hayward, CA, pp. 120–141.Google Scholar
Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts (Monogr. Stat. Appl. Prob. 73). Chapman and Hall, London.CrossRefGoogle Scholar
Joe, H., Li, H. and Nikoloulopoulos, A. K. (2010). Tail dependence functions and vine copulas. J. Multivariate Anal. 101, 252–270.CrossRefGoogle Scholar
Kano, Y. (1994). Consistency property of elliptic probability density functions. J. Multivariate Anal. 51, 139–147.CrossRefGoogle Scholar
Kellerer, H. G. (1964). Verteilungsfunktionen mit gegebenen marginalverteilungen Z. Wahrschein lichkeitsth. 3, 247–270.CrossRefGoogle Scholar
Li, H., Scarsini, M. and Shaked, M. (1996). Linkages: A tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. J. Multivariate Anal. 56, 20–41.CrossRefGoogle Scholar
Li, H., Scarsini, M. and Shaked, M. (1999). Dynamic linkages for multivariate distributions with given nonoverlapping multivariate marginals. J. Multivariate Anal. 68, 54–77.CrossRefGoogle Scholar
Marco, J. M. and Ruiz-Rivas, C. (1992). On the construction of multivariate distributions with given nonoverlapping multivariate marginals. Statist. Prob. Lett. 15, 259–265.CrossRefGoogle Scholar
Maronna, R., Martin, D. and Yohai, V. (2006). Robust Statistics: Theory and Methods. John Wiley, Chichester.CrossRefGoogle Scholar
McNeil, A. J. (2008). Sampling nested archimedean copulas. J. Statist. Comput. Simul. 78, 567–581.CrossRefGoogle Scholar
McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and $\ell_1$-norm symmetric distributions. Ann. Statist . 37, 3059–3097.CrossRefGoogle Scholar
Nelsen, R. B. (1999). An Introduction to Copulas (Lecture Notes Statist. 139). Springer, New York.CrossRefGoogle Scholar
Nelsen, R. B. (2005). Some properties of schur-constant survival models and their copulas. Braz. J. Prob. Statist. 19, 179–190.Google Scholar
Portnoy, S. (2003). Censored regression quantiles. J. Amer. Statist. Assoc. 98, 1001–1012.CrossRefGoogle Scholar
Rüschendorf, L. (1985). Construction of multivariate distributions with given marginals. Ann. Inst. Statist. Math. 37, 225–233.CrossRefGoogle Scholar
Sánchez Algarra, P. (1986). Construcción de distribuciones con marginales multivariantes dadas. Qüestiió 10, 133–141.Google Scholar
Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Google Scholar
Spanier, E. H. (1994). Algebraic Topology. Springer, New York.Google Scholar