Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 975–1001 | Cite as

Dependence Properties of Conditional Distributions of some Copula Models

  • Harry Joe


For multivariate data from an observational study, inferences of interest can include conditional probabilities or quantiles for one variable given other variables. For statistical modeling, one could fit a parametric multivariate model, such as a vine copula, to the data and then use the model-based conditional distributions for further inference. Some results are derived for properties of conditional distributions under different positive dependence assumptions for some copula-based models. The multivariate version of the stochastically increasing ordering of conditional distributions is introduced for this purpose. Results are explained in the context of multivariate Gaussian distributions, as properties for Gaussian distributions can help to understand the properties of copula extensions based on vines.


Factor model Markov tree Mixture of conditional distributions Positive dependence Stochastically increasing Total positivity of order 2 Vine 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. To Begin With, Silver Spring, MDGoogle Scholar
  2. Bedford T, Cooke RM (2001) Probability density decomposition for conditionally dependent random variables modeled by vines. Ann Math Artif Intell 32(1-4):245–268MathSciNetCrossRefMATHGoogle Scholar
  3. Bedford T, Cooke RM (2002) Vines — a new graphical model for dependent random variables. Ann Stat 30(4):1031–1068MathSciNetCrossRefMATHGoogle Scholar
  4. Bernard C, Czado C (2015) Conditional quantiles and tail dependence. J Multivar Anal 138(SI):104–126MathSciNetCrossRefMATHGoogle Scholar
  5. Bouyé E, Salmon M (2009) Dynamic copula quantile regressions and tail area dynamic dependence in Forex markets. European Journal of Finance 15(7-8):721–750CrossRefGoogle Scholar
  6. Brechmann EC, Czado C, Aas K (2012) Truncated regular vines in high dimensions with application to financial data. Can J Stat 40(1):68–85MathSciNetCrossRefMATHGoogle Scholar
  7. Chen X, Koenker R, Xiao Z (2009) Copula-based nonlinear quantile autoregression. Econ J 12(1):S50–S67MathSciNetMATHGoogle Scholar
  8. Cooke R, Joe H, Chang B (2015) Vine regression. Technical report, Resources for the Future, RFF DP 15–52Google Scholar
  9. Fang Z, Joe H (1992) Further developments on some dependence orderings for continuous bivariate distributions. Ann Inst Stat Math 44(3):501–517MathSciNetMATHGoogle Scholar
  10. Hua L, Joe H (2011) Tail order and intermediate tail dependence of multivariate copulas. J Multivar Anal 102(10):1454–1471MathSciNetCrossRefMATHGoogle Scholar
  11. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, LondonCrossRefMATHGoogle Scholar
  12. Joe H (2014) Dependence Modeling with Copulas. Chapman & Hall/CRC, Boca Raton, FLMATHGoogle Scholar
  13. Karlin S, Rinott Y (1980) Classes of orderings of measures and related correlation inequalities i. Multivariate totally positive distributions. J Multivar Anal 10:467–498MathSciNetCrossRefMATHGoogle Scholar
  14. Kendall M, Stuart A (1977) The advanced theory of statistics, vol 2, 4th edn. Charles Griffin & Co. Ltd., LondonGoogle Scholar
  15. Kraus D, Czado C (2016) D-vine copula based quantile regression. Technical report, Technische Universität MünchenGoogle Scholar
  16. Krupskii P, Joe H (2013) Factor copula models for multivariate data. J Multivar Anal 120:85–101MathSciNetCrossRefMATHGoogle Scholar
  17. Marshall AW, Olkin I (1988) Families of multivariate distributions. J Am Stat Assoc 83(403):834–841MathSciNetCrossRefMATHGoogle Scholar
  18. Marshall AW, Olkin I (2007) Life distributions, structure of nonparametric, semiparametric, and parametric families. Springer Series in Statistics. Springer, New YorkMATHGoogle Scholar
  19. Müller A, Scarsini M (2001) Stochastic comparison of random vectors with a common copula. Math Oper Res 26(4):723–740MathSciNetCrossRefMATHGoogle Scholar
  20. Müller A, Scarsini M (2005) Archimedean copulae and positive dependence. J Multivar Anal 93(2):434–445MathSciNetCrossRefMATHGoogle Scholar
  21. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkCrossRefMATHGoogle Scholar
  22. Whittaker J (1990) Graphical models in applied multivariate statistics. Wiley, ChichesterMATHGoogle Scholar
  23. Yanagimoto T, Okamoto M (1969) Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann Inst Stat Math 21(3):489–506MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of British ColumbiaVancouverCanada

Personalised recommendations