Copula (probability theory)

A copula (Pl. copula or copula ) is a function that may indicate a functional relationship between the marginal distribution functions of several random variables and their joint probability distribution.

With their help, you can model stochastic dependence significantly more flexible than, for example, with correlation coefficients.

Definition

A copula is a multivariate distribution function whose one-dimensional marginal distributions uniformly distributed over the interval are. More formally, this means the following:

• Is multivariate distribution function, that is,,
• Is n- increasing, that is, for each hyper- rectangle is the C volume is not negative :, where,

• The one-dimensional marginal distributions of are uniform on the unit interval.

Requiring the marginal distributions can be motivated as follows: For any random variables with continuous distributions the random variable is uniformly distributed over the interval. Together with the following set of Sklar separating marginal distributions and dependencies among these is possible.

Set of Sklar

In the following, an extension of the real numbers.

Be one -dimensional distribution function with one-dimensional marginal distributions. Then there exists a -dimensional copula, so that applies to all:

Are all continuous, then the copula is unique.

Fréchet - Hoeffding barriers

For each - variate copula the lower Fréchet - Hoeffding bound applies

And the upper Fréchet - Hoeffding bound

The upper bound is itself a copula, the lower bound on the other hand only.

Thus it can be concluded, among other things:

• For all

Application

Copulas are used to obtain inferences on the nature of the stochastic dependence of various random variables or to model dependencies targeted. They are used for example in credit risk analysis in order to make statements about a heaping several bankruptcy debtor within a bond portfolio can. Applications are analog common in the insurance sector. There heaped provide damages of various types of damage a financial problem dar. example is an observable relationship between storm and flood damage.

Examples of copulas

• The simplest form of the copula is the Unabhängigkeitscopula ( Produktcopula )

• The upper Fréchet - Hoeffding barrier, also a copula is given by

• The lower Fréchet - Hoeffding barrier is only in the bivariate case, a copula:

• The normal or Gaussian copula is defined by means of the distribution function of the normal distribution. so is

• The Gumbel copula is defined by means of the exponential function and the natural logarithm

Archimedean copulas

Archimedean copulas are a class of copulas dar. This can be described as follows:

Let be a continuous, monotonically decreasing function with. Denote the pseudo - inverse of, ie

With the help of and it is now possible to define a bivariate function:

The function is exactly then a copula if it is convex. In this case, ie producers of the copula. Obviously is symmetric, that is, for all.

Examples of Archimedean copulas are:

• Gumbel copula: Your generator is the function with parameters.

• Clayton copula: Your generator is the function with parameters.

• Frank copula: Your generator is the function with parameters.

Archimedean copulas are often used because it is very easy to generate random numbers from it.

Extremwertcopula

Definition

A copula is, Extremwertcopula if it is the copula of a multivariate extreme value distribution, ie there is a multivariate extreme value distribution with univariate margins that apply.

Lemma

A copula is exactly then a Extremwertcopula if for and.

Is a Extremwertcopula and are univariate extreme value distributions, then a multivariate extreme value distribution.