The Use of Copulas To Model Conditional Expectation for Multivariate Data
Meelis Käärik, Anne Selart, Ene Käärik
Institute of Mathematical Statistics, University of Tartu, Tartu, Estonia

In this paper we generalize a copula based conditional expectation model for correlated multivariate data (Käärik, Käärik, 2009, 2010). There are two main statistical advantages of modeling the dependence of multivariate data by copulas. Firstly, copulas allow us to use arbitrary marginals (that even need not to come from same distributional family). Secondly, the copula technique allows us to separate the modeling of marginals from modeling of the dependence structure.

In our previous studies we focused on the Gaussian copula case, which is a natural starting point, in current research we expand the class of copulas, including t-copula, and also go beyond the elliptical class of copulas, examining few related skewed copulas (e.g. skew-normal) for modeling asymmetric data.

Our main aim is to generalize the obtained results for Gaussian copula (including joint and conditional distribution functions and conditional expectations for different dependence structures and related imputation algorithms) to a broader class of copulas. The approach with different choices of copulas naturally raises the question which copula model suits best to our problem. To identify the appropriate copula we can apply certain goodness of fit tests (besides the graphical methods). We refer to recent works by Berg (2009) and Kojadinovic and Yan (2010) on that subject.

Although initially designed for the imputation problem of missing data, the setup allows wide implementation in many fields, including insurance (especially credibility models) and microarray data.

This work is supported by Estonian Science Foundation grants No 7313 and No 8294.


Berg, D. (2009). Copula Goodness-of-Fit Testing: An Overview and Power Comparison. The European Journal of Finance, 15, 675 - 701.

Käärik, M.; Käärik, E. (2010). Imputation by Gaussian Copula Model with an Application to Incomplete Customer Satisfaction Data. Lechevallier, Y.; Saporta, G. (Ed.). Proceedings of Compstat'2010: 19th International Conference on Computational Statistics, 485 - 492. Springer-Verlag,

Käärik, E., Käärik, M. (2009). Modelling Dropouts by Conditional Distribution, a Copula-Based Approach. Journal of Statistical Planning and Inference, 139(11), 3830 - 3835.

Yan, J., Kojadinovic, I (2010). Modeling Multivariate Distributions with Continuous Margins Using the copula R Package. Journal of Statistical Software, Vol 34, 9, 1 - 20.

Keywords: t-Copula; Skew-normal Copula; Imputation

Biography: Studies: 2005 – PhD in Mathematical Statistics, University of Tartu, Estonia

Academic positions: 2008–present – Senior Researcher, Institute of Math. Statistics, University of Tartu, Estonia

Main research fields: Approximation of distributions, Risk models in non-life insurance, Analysis of missing data, imputation