Some copulas

The word 'copula' originates from the Latin noun for a "link or tie" that connects two different things. Words such as 'couple' or even 'copulate' stem from the same source.

In Mathematics, copulas are used to couple two or more univariate distributions. The univariate distributions can be specified in isolation, and then coupled afterwards. For example can a copula join together a Pareto distribution and a Gaussian distribution, or two Student-t distributions with various degrees of freedom. This kind of flexibility is not found in traditional multivariate statistics. For example, in a mulitvariate Student-t distribution, all marginal distributions must share the same tail heavyness.

More formally, Wolfram describes copulas as follows:

Let H be a two-dimensional distribution function with marginal distribution functions F and G. Then there exists a copula C such that H(x,y)=C(F(x),G(y)).

Conversely, for any univariate distribution functions F and G and any copula C, the function H is a two-dimensional distribution function with marginals F and G. Furthermore, if F and G are continuous, then C is unique.

Weisstein, Eric W. "Sklar's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SklarsTheorem.html

There exists a lot of Copulas, among which the Gaussian and Student-t are the most popular. These two, so called elliptical copulas, are symmetrical in that they deliver equal probabilities for both high and low joint movements.

One significant difference exists though. A Gaussian copula has zero probability for joint extreme movements, while a Student-t distribution has non-zero probability for joint extreme movements. In for example a financial application, if the value of one stock hits rock bottom, only the Student-t copula allows the other stock to plunge as well. Therefore, in risk management, where everything that can go wrong, will go wrong at the same time, the student-t copula should be prefered.