Copula functions

June 24, 2015 — June 24, 2015

Figure 1

A neat way of quantifying dependence structures between random variables. Useful in, e.g. Quantitative Risk Management.

The trick is simple: Informally, you look at the marginal iCDF of each of \(n\) variables, and fiddle with the joint distribution of those marginals on \([0,1]^n\). (That’s assuming variables are absolutely continuous w.r.t some underlying measure space; distribution with atoms are more fiddly.)

This is a good trick, although I need to sit down and think it through. I would like to better understand:

1 Elliptical

For now, see elliptical distributions.

2 Vine copulas

Hierarchical graphical models, AFAICS.

3 References

Bielecki, Jakubowski, and Niewęgłowski. 2020. Fundamentals of the Theory of Structured Dependence Between Stochastic Processes. Encyclopedia of Mathematics and Its Applications.
Chen, and Huang. 2007. Nonparametric Estimation of Copula Functions for Dependence Modelling.” Canadian Journal of Statistics.
Dette, Siburg, and Stoimenov. 2013. A Copula-Based Non-Parametric Measure of Regression Dependence.” Scandinavian Journal of Statistics.
Embrechts. 2009. Copulas: A Personal View.” Journal of Risk and Insurance.
Embrechts, Lindskog, and McNeil. 2003. Modelling Dependence with Copulas and Applications to Risk Management.” Handbook of Heavy Tailed Distributions in Finance.
Embrechts, McNeil, and Straumann. 2002. “Correlation and Dependence in Risk Management: Properties and Pitfalls.” Risk Management: Value at Risk and Beyond.
Fan, and Patton. 2014. Copulas in Econometrics.” Annual Review of Economics.
Genest, and Favre. 2007. Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask.” Journal of Hydrologic Engineering.
Härdle, and Okhrin. 2010. De Copulis Non Est Disputandum.” Advances in Statistical Analysis.
Hofert. 2008. Sampling Archimedean Copulas.” Computational Statistics & Data Analysis.
Hofert, and Vrins. 2013. Sibuya Copulas.” Journal of Multivariate Analysis.
Klein, Nott, and Smith. 2021. Marginally Calibrated Deep Distributional Regression.” Journal of Computational and Graphical Statistics.
Klein, and Smith. 2019. Implicit Copulas from Bayesian Regularized Regression Smoothers.” Bayesian Analysis.
Kraskov, Stögbauer, and Grassberger. 2004. Estimating Mutual Information.” Physical Review E.
MacKenzie, and Spears. 2014. ‘The Formula That Killed Wall Street’: The Gaussian Copula and Modelling Practices in Investment Banking.” Social Studies of Science.
Mai, and Scherer. 2011. Reparameterizing Marshall–Olkin Copulas with Applications to Sampling.” Journal of Statistical Computation and Simulation.
———. 2012. Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. Series in Quantitative Finance, v. 4.
Malevergne, and Sornette. 2003. Testing the Gaussian Copula Hypothesis for Financial Assets Dependences.” Quantitative Finance.
Ma, and Sun. 2011. Mutual Information Is Copula Entropy.” Structural Change and Economic Dynamics.
McNeil. 2008. Sampling Nested Archimedean Copulas.” Journal of Statistical Computation and Simulation.
Nelsen. 1999. An Introduction to Copulas. Lecture Notes in Statistics 139.
Patton, A J. 2009. “Copula-Based Models for Financial Time Series.” In Handbook of Financial Time Series.
Patton, Andrew J. 2012. A Review of Copula Models for Economic Time Series.” Journal of Multivariate Analysis.
Rémillard, Papageorgiou, and Soustra. 2012. Copula-Based Semiparametric Models for Multivariate Time Series.” Journal of Multivariate Analysis.
Schmidt. 2006. “Coping with Copulas.” In Copulas from Theory to Applications in Finance.
Shaw. 2006. Sampling Student’s T Distribution – Use of the Inverse Cumulative Distribution Function.” Journal of Computational Finance.
Trivedi, and Zimmer. 2006. Copula Modeling: An Introduction for Practitioners.” Foundations and Trends® in Econometrics.
Wang, and Yin. 2020. Relaxed Multivariate Bernoulli Distribution and Its Applications to Deep Generative Models.” In Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI).
Watts. 2016. “The Gaussian Copula and the Financial Crisis: A Recipe for Disaster or Cooking the Books?”
Whelan. 2004. Sampling from Archimedean Copulas.” Quantitative Finance.
Wu, Valdez, and Sherris. 2007. Simulating from Exchangeable Archimedean Copulas.” Communications in Statistics - Simulation and Computation.