In statistics, a copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 1]ⁿ such that every marginal distribution is uniform on the interval [0, 1]. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... A unit cube is a 3-dimensional geometric figure that consists of a cube in which all of its dimensions are 1 unit long. ... In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ...

Sklar's theorem is 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

(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. The copula contains all of the information on the nature of the dependence between the two random variables that can be given without the marginal distributions, but gives no information on the marginal distributions. In effect the information on the marginals and the information on the dependence are neatly separated from each other. In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

Some copula properties are the following:

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.

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

Thus, for all copulas C(u,v),

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 Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ... The normal distribution, also called Gaussian distribution (although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ...

where the marginals of X and Y are N(0,1) distributions and Φ denotes the cumulative normal density. Differentiating this yields

where

is the density function for the bivariate normal variate with Pearson's product moment correlation coefficient ρ, φ is the density of the N(0,1) distribution (the marginal density).

Archimedean copulas

One particularly simple form of a copula is

where ψ 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: In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ...

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

Clayton copula:

For θ = 1 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. In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. ...

References

• 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. JSTOR (subscription)
• Frees, E.W., Valdez, E.A. (1998), "Understanding Relationships Using Copulas", North American Actuarial Journal 2, 1-25. 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 leures marges", Publications de l'Institut de Statistique de L'Université de Paris 8, 229-231.

External links

• MathWorld Eric W. Weisstein. "Sklar's Theorem." From MathWorld--A Wolfram Web Resource