A neat way of quantifying arbitrary (?) 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 variables, and fiddle with the joint distribution of those marginals on . (That's assuming variables are absolutely continuous w.r.t some underlying measure space; distribution with atoms are more tricky.)
This is a good trick, although I need to sit down and think through it. I would like to better understand:
the relationship between the underlying event space and the instrumental one we “sort of” construct in copula modeling.
Is any information lost with non-monotonic coupling in a copula model?
conditional copulas and how they work
the occasionally-mentioned relationship between copula entropy and mutual information.
- Patt12: Andrew J Patton (2012) A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18. DOI
- Nels99: Roger B. Nelsen (1999) An introduction to copulas. New York: Springer
- Muli18: Sabrina Mulinacci (2018) Archimedean-based Marshall-Olkin Distributions and Related Copula Functions. Methodology and Computing in Applied Probability, 20(1), 205–236. DOI
- Schm06: Thorsten Schmidt (2006) Coping with copulas. In Copulas from theory to applications in finance.
- TrZi06: Pravin K Trivedi, David M Zimmer (2006) Copula Modeling: An Introduction for Practitioners. Foundations and Trends® in Econometrics, 1(1), 1–111. DOI
- Patt09: A J Patton (2009) Copula-based models for financial time series. In Handbook of financial time series (pp. 767–785). Berlin, Heidelberg: Springer Berlin Heidelberg
- RéPS12: Bruno Rémillard, Nicolas Papageorgiou, Frédéric Soustra (2012) Copula-based semiparametric models for multivariate time series. Journal of Multivariate Analysis, 110, 30–42. DOI
- Embr09: Paul Embrechts (2009) Copulas: A Personal View. Journal of Risk and Insurance, 76(3), 639–650. DOI
- FaPa14: Yanqin Fan, Andrew J. Patton (2014) Copulas in Econometrics. Annual Review of Economics, 6(1), 179–200. DOI
- EmMS02: Paul Embrechts, Alexander J McNeil, Daniel Straumann (2002) Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and Beyond, 176–223.
- HäOk10: Wolfgang Härdle, Ostap Okhrin (2010) De copulis non est disputandum. Advances in Statistical Analysis, 94(1), 1–31. DOI
- Yan07: J Yan (2007) Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
- KrSG04: Alexander Kraskov, Harald Stögbauer, Peter Grassberger (2004) Estimating mutual information. Physical Review E, 69, 066138. DOI
- ChFa06: Xiaohong Chen, Yanqin Fan (2006) Estimation of copula-based semiparametric time series models. Journal of Econometrics, 130(2), 307–335. DOI
- GeFa07: C Genest, A C Favre (2007) Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, 347. DOI
- UyMa16: Nathan Uyttendaele, Gildas Mazo (2016) Extending one-factor copulas. ArXiv:1612.02848 [Stat].
- EmLM03: Paul Embrechts, Filip Lindskog, Alexander J McNeil (2003) Modelling dependence with copulas and applications to risk management. Handbook of Heavy Tailed Distributions in Finance, 8(329–384), 1.
- MaSu11: Jian Ma, Zengqi Sun (2011) Mutual Information Is Copula Entropy. Structural Change and Economic Dynamics, 16(1), 51–54. DOI
- ChHu07: Song Xi Chen, Tzee-Ming Huang (2007) Nonparametric estimation of copula functions for dependence modelling. Canadian Journal of Statistics, 35(2), 265–282. DOI
- ScFe02: Olivier Scaillet, Jean-David Fermanian (2002) Nonparametric Estimation of Copulas for Time Series (SSRN Scholarly Paper No. ID 372142). Rochester, NY: Social Science Research Network
- MaSc11: Jan-Frederik Mai, Matthias Scherer (2011) Reparameterizing Marshall–Olkin copulas with applications to sampling. Journal of Statistical Computation and Simulation, 81(1), 59–78. DOI
- Hofe08: Marius Hofert (2008) Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52(12), 5163–5174. DOI
- Whel04: Niall Whelan (2004) Sampling from archimedean copulas. Quantitative Finance, 4, 339–352. DOI
- Mcne08: Alexander J. McNeil (2008) Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581. DOI
- Shaw06: William T Shaw (2006) Sampling Student’s T distribution – use of the inverse cumulative distribution function. Journal of Computational Finance.
- HoVr13: Marius Hofert, Frédéric Vrins (2013) Sibuya copulas. Journal of Multivariate Analysis, 114, 318–337. DOI
- MaSc12: Jan-Frederik Mai, Matthias Scherer (2012) Simulating copulas: stochastic models, sampling algorithms, and applications. London : Hackensack, NJ: Imperial College Press ; World Scientific
- WuVS07: Florence Wu, Emiliano Valdez, Michael Sherris (2007) Simulating from Exchangeable Archimedean Copulas. Communications in Statistics - Simulation and Computation, 36(5), 1019–1034. DOI
- MaSo03: Yannick Malevergne, Didier Sornette (2003) Testing the Gaussian copula hypothesis for financial assets dependences. Quantitative Finance, 3(4), 231–250. DOI