Copula (statistics)

In statistics, a copula is used as a general way of formulating a multivariate distribution in such a way that various general types of dependence can be represented. Other ways of formulating multivariate distributions include conceptually-based approaches in which the real-world meaning of the variables is used to imply what types of relationships might occur. In contrast, the approach via copulas might be considered as being more raw, but it does allow much more general types of dependencies to be included than would usually be invoked by a conceptual approach.

The approach to formulating a multivariate distribution using a copula is based on the idea that a simple transformation can be made of each marginal variable in such a way that each transformed marginal variable has a uniform distribution. When applied in a practical context, such transformations might be fitted as an initial step for each margin, or the parameters of the transformations might be fitted jointly with those of the copula.

There are many families of copulas which differ in the detail of the dependence they represent. A family will typically have several parameters which relate to the strength and form of the dependence. However, it is possible to specify a dependency structure and for a copula to emerge using a conditioning technique such as the D distribution.

Definition

A copula is a multivariate joint distribution defined on the "n"-dimensional unit cube [0, 1] ^"n" such that every marginal distribution is uniform on the interval [0, 1] .

Specifically, $C:\; [0,1]\; ^n\; o\; [0,1]$ is an "n"-dimensional copula (briefly, "n"-copula) if:

: $Cleft(mathbf\; u\; ight)=0$ whenever $mathbf\; uin\; [0,1]\; ^n$ has at least one component equal to $0;$

: $Cleft(mathbf\; u\; ight)=u\_i$ whenever $mathbf\; uin\; [0,1]\; ^n$ has all the components equal to $1$ except the i-th one, which is equal to $u\_i;$

: $Cleft(mathbf\; u\; ight)$ is n-increasing, i.e., for each $B=\; imes\_\{i=1\}^\{n\}\; [x\_i,y\_i]\; subseteq\; [0,1]\; ^n;$

: $V\_\{C\}left(\; B\; ight):=sum\_\{mathbf\; zin\; imes\_\{i=1\}^\{n\}\{x\_i,y\_i\; (-1)^\{N(mathbf\; z)\}\; C(mathbf\; z)ge\; 0;$

where the $N(mathbf\; z)=operatorname\{card\}\{kmid\; z\_k=x\_k\}$.$V\_\{C\}left(\; B\; ight)$ is the so called $C$-volume of $B$.

klar's theorem

The theorem proposed by Sklar Sklar (1959)] underlies most applications of the copula. Sklar's theorem states that given a joint distribution function "H" for "p" variables, and respective marginal distribution functions, there exists a copula "C" such that the copula binds the margins to give the joint distribution.

For the bivariate case, Sklar's theorem can be stated as follows. For any bivariate distribution function "H"("x", "y"), let "F"("x") = "H"("x", (−∞,∞)) and "G"("y") = "H"((−∞,∞), "y") be the univariate marginal probability distribution functions. Then there exists a copula "C" such that

:$H(x,y)=C(F(x),G(y)),$

(where we have identified the distribution "C" with its cumulative distribution function). Moreover, if marginal distributions, say, "F"("x") and "G"("y"), are continuous, the copula function "C" is unique. Otherwise, the copula "C" is unique on the range of values of the marginal distributions.

Fréchet-Hoeffding copula boundaries

Minimum copula: This is the lower bound for all copulas. In the bivariate case only, it represents perfect negative dependence between variates.

:$W(u,v)\; =\; max(0,u+v-1).,$

For $n$-variate copulas, the lower bound is given by:$W(u\_1,ldots,u\_n)\; :=\; maxleft\{1-n+sumlimits\_\{i=1\}^n\; \{u\_i\}\; ,\; 0\; ight\}\; leq\; C(u\_1,ldots,u\_n).$

Maximum copula: This is the upper bound for all copulas. It represents perfect positive dependence between variates:

:$M(u,v)\; =\; min(u,v).,$

For $n$-variate copulas, the upper bound is given by:$C(u\_1,ldots,u\_n)le\; min\_\{j\; in\; \{1,ldots,n\; u\_j\; =:\; M(u\_1,ldots,u\_n).$

Conclusion: For all copulas $C(u,v)$,:$W(u,v)\; le\; C(u,v)\; le\; M(u,v).$

In the multivariate case, the corresponding inequality is:$W(u\_1,ldots,u\_n)\; le\; C(u\_1,ldots,u\_n)\; le\; M(u\_1,ldots,u\_n).$

Gaussian copula

One example of a copula often used for modelling in finance is the Gaussian Copula, which is constructed from the bivariate normal distribution via Sklar's theorem. For "X" and "Y" distributed as standard bivariate normal with correlation "ρ" the Gaussian copula function is

:$C\_\; ho(U,V)\; =\; Phi\_\{\; ho\}\; left(Phi^\{-1\}(U),\; Phi^\{-1\}(V)\; ight)$

where U = U(X) and V = V(Y) are cumulative probability distributions whose result is in (0,1) and Φ denotes the cumulative normal density. These are "percentile to percentile transfomations", where X and Y are transformed into

:$A\; =\; (Phi^\{-1\}(U))$ and $B\; =(Phi^\{-1\}(V))$

Differentiating C yields to:

:$c\_\; ho(U,V)\; =\; frac\{Phi\_\{U,V,\; ho\}\; (Phi^\{-1\}(U),\; Phi^\{-1\}(V)\; )\}\{Phi(Phi^\{-1\}(U))\; Phi(Phi^\{-1\}(V))\}$

where:

:$Phi\_\{X,Y,\; ho\}(x,y)\; =\; frac\{1\}\{2\; pisqrt\{1-\; ho^2\; exp\; left\; (-frac\{1\}\{2(1-\; ho^2)\}\; left\; [\{x^2+y^2\}\; -2\; ho\; xy\; ight\; ]\; ight\; )$

is the density function for the standard bivariate gaussian with Pearson's product moment correlation coefficient $ho$.

Archimedean copulas

Archimedean copulas are an important family of copulas, which have a simple form with properties such as associativity and have a variety of dependence structures. Unlike elliptical copulas (eg. Gaussian), most of the Archimedean copulas have closed-form solutions and are not derived from the multivariate distribution functions using Sklar’s Theorem.

One particularly simple form of a "n"-dimensional copula is

:$H(x\_1,x\_2,dots,x\_n)\; =\; Psi^\{-1\}left(sum\_\{i=1\}^nPsi(F\_i(x\_i))\; ight),$

where $Psi$ is known as a "generator function". Such copulas are known as "Archimedean". Any generator function which satisfies the properties below is the basis for a valid copula:

:

Product copula: Also called the independent copula, this copula has no dependence between variates. Its density function is unity everywhere.

:$Psi(x)\; =\; -ln(x);\; qquad\; H(x,y)\; =\; F(x)G(y).$

Where the generator function is indexed by a parameter, a whole family of copulas may be Archimedean. For example:

Clayton copula:

:$Psi(x)\; =\; x^\{\; heta\}\; -1;qquad\; heta\; le\; 0;\; qquad\; H(x,y)\; =\; (F(x)^\; heta+G(y)^\; heta-1)^\{1/\; heta\}.$

For θ = 0 in the Clayton copula, the random variables are statistically independent. The generator function approach can be extended to create multivariate copulas, by simply including more additive terms.

Gumbel copula::$Psi(x)=\; (-ln(x))^alpha\; ,.$

Frank copula::$Psi(x)=\; lnleft(\; frac\{e^\{alpha\; x\}\; -1\}\{e^\{alpha\}\; -1\}\; ight)\; .$

Applications

Copulas are used in the pricing of collateralized debt obligations.Fact|date=October 2008 Dependence modelling with copula functions is widelyused in applications of financial risk assessment and actuarial analysis. Recently they have been successfully applied to the database formulation for the reliability analysis of Highway bridges and to various multivariate simulation studies in Civil, Mechanical and Offshore engineering.Fact|date=October 2008

ee also

*D distribution

References

Notes

General

* David G. Clayton (1978), "A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence", "Biometrika" 65, 141-151. [http://links.jstor.org/sici?sici=0006-3444%28197804%2965%3A1%3C141%3AAMFAIB%3E2.0.CO%3B2-Y JSTOR (subscription)]
* Frees, E.W., Valdez, E.A. (1998), "Understanding Relationships Using Copulas", "North American Actuarial Journal" 2, 1-25. [http://www.soa.org/library/journals/north-american-actuarial-journal/1998/january/naaj9801_1.pdf Link to NAAJ copy]
* Roger B. Nelsen (1999), "An Introduction to Copulas". ISBN 0-387-98623-5.
* S. Rachev, C. Menn, F. Fabozzi (2005), "Fat-Tailed and Skewed Asset Return Distributions". ISBN 0-471-71886-6.
* A. Sklar (1959), "Fonctions de répartition à "n" dimensions et leurs marges", "Publications de l'Institut de Statistique de L'Université de Paris" 8, 229-231.
* C. Schölzel, P. Friederichs (2008), "Multivariate non-normally distributed random variables in climate research – introduction to the copula approach". [http://www.meteo.uni-bonn.de/mitarbeiter/CSchoelzel/sources/schoelzel_egu_2008_copulas_talk_01.pdf PDF]
* W.T. Shaw, K.T.A. Lee (2006), "Copula Methods vs Canonical Multivariate Distributions: The Multivariate Student T Distibution with General Degrees of Freedom". [http://www.mth.kcl.ac.uk/~shaww/web_page/papers/MultiStudentc.pdf PDF]
* Srinivas Sriramula, Devdas Menon and A. Meher Prasad (2006), "Multivariate Simulation and Multimodal Dependence Modeling of Vehicle Axle Weights with Copulas", "ASCE Journal of Transportation Engineering" 132 (12), 945–955. (doi 10.1061/(ASCE)0733-947X(2006)132:12(945)) [http://cedb.asce.org/cgi/WWWdisplay.cgi?0613154 ASCE(subscription)]

External links

* [http://mathworld.wolfram.com/SklarsTheorem.html MathWorld Eric W. Weisstein. "Sklar's Theorem." From MathWorld--A Wolfram Web Resource]
* [http://www.flll.jku.at/mediawiki Copula Wiki: community portal for researchers with interest in copulas]
* [http://www.vosesoftware.com/ModelRiskHelp/index.htm#Modeling_correlation/Archimedean_copulas_-_the_Clayton_Frank_and_Gumbel.htm Explanation of Archimedean copulas: The Clayton, Frank and Gumbel]
* [http://www.mathfinance.cn/tags/copula A collection of Copula simulation and estimation codes]

oftware

* [http://www.vosesoftware.com/ModelRiskHelp/index.htm#Help_on_ModelRisk/Copulas/Copulas_in_ModelRisk.htm A tool for modeling copulas in Excel] by Vose Software