Extreme Value Copulas

Extreme value copulas are fundamental in the study of rare and extreme events due to their ability to model dependency in situations of extreme risk. This package proposes a wide selection of bivariate extreme values copulas, while multivariate cases are not implemented yet. Feel free to open an issue and/or propose pull requests if you want an implementation of a multivariate case.

A bivariate extreme value copula [34] $C$ is a copula that has the following caracteristic property:

\[C(u_1^t, u_2^t)=C(u_1,u_2)^t, t > 0.\]

It can be represented through its stable tail dependence function $\ell(\cdot)$:

\[C(u_1, u_2)=\exp\{-\ell(\log(u_1),\log(u_2))\},\]

or through a convex function $A: [0,1] \to [1/2, 1]$ satisfying $\max(t, t-1)\leq A(t) \leq 1,$ called its Pickands dependence function:

\[C(u_1,u_2)=\exp\left\{\log(u_1u_2)A\left(\frac{\log(u_1)}{\log(u_1u_2)}\right)\right\},\]

In the context of bivariate extreme value copulas, the functions $\ell$ and $A$ are related as follows:

\[\ell(u_1,u_2)=(u_1+u_2)A\left(\frac{u_1}{u_1+u_2}\right).\]

Therefore, in this implementation, it is sufficient to provide the Pickands dependence function $A$ to construct the implementation structure of an extreme value copula.

In this package, there is an abstract type ExtremeValueCopula that provides a foundation for defining extreme value copulas. Many extreme value copulas are already implemented for you! See this list to get an overview.

If you do not find the one you need, you may define it yourself by subtyping ExtremeValueCopula. The API does not require much information, which is really convenient. Only a method for the pickand function A(C::ExtremeValueCopula) = ... is required. By providing this functions, you can easily create a new extreme value copula that fits your specific needs:

struct MyExtremeValueCopula{P} <: ExtremeValueCopula{P}
    θ::P
end

A(C::ExtremeValueCopula, t) = (t^C.θ + (1 - t)^C.θ)^(1/C.θ) # This is the Pickands function of the Logistic (Gumbel) Copula

Advanced Concepts

Here, we present some important concepts from the theory of extreme value copulas that are useful for the development of this package.

Let $(X,Y) \sim C$ where $C$ is a bivariate extreme value copula. We have the following results:

Proposition 1 (Ghoudi 1998 [35]): Let $(X, Y) \sim C$, where $C$ is an extreme value copula. The joint distribution of $X$ and $Z = \frac{\log(X)}{\log(XY)}$ is given by:

\[P(Z \leq z, X \leq x) =G(z,x)=\left(z + z(1-z)\frac{A'(z)}{A(z)}\right)x^{A(z)/z}, \quad 0\leq x,z \leq 1\]

where $A'(z)$ denotes the derivate of function $A(z)$ at point $z.$

Since $A$ is a convex function defined on $[0, 1]$ and satisfies $-1 \leq A'(z) \leq 1$, by extension, we define $A'(1)$ as the supremum of $A'(z)$ over $(0, 1)$. By setting $x = 1$ in the previous result, we obtain the marginal distribution of $Z$: $P(Z \leq z) = G_Z(z) = z + z(1 - z) \frac{A'(z)}{A(z)}, \quad 0 \leq z \leq 1.$

This result was demonstrated by Deheuvels (1991) [36] in the case where $A$ admits a second derivative.

Simulation of Bivariate Extreme Value Distributions

To simulate a bivariate extreme value distribution $C(x, y)$, remark that if $F_1$ and $F_2$ are univariate extreme value distributions, then the pair $( F_1^{-1}(X), F_2^{-1}(Y) )$ is distributed according to a bivariate extreme value distribution. The proposed algorithm in Ghoudi,1998, [35] allows simulating such a distribution.

Assume $A$ has a second derivative, making the distribution absolutely continuous. In this case, $Z$ is also absolutely continuous and has a density $g_Z(z)$ given by:

\[g_Z(z) = \frac{d}{dz} G_Z(z) = 1 + (1 - z)^{-1} \left(A(z) - z A'(z)\right)\]

The conditional distribution of $W$ given $Z$ is:

\[F(w|z) = \frac{1}{g_Z(z)} \frac{d}{dz} F(z, w),\]

which simplifies to:

\[F(w|z) = w \frac{z(1 - z) A'(z)}{A(z) g_Z(z)} + (w - w \log w) \left(1 - \frac{z(1 - z) A''(z)}{A(z) g_Z(z)} \right)\]

Given $Z$, the distribution of $W$ is uniform on $(0, 1)$ with probability $p(Z)$ and equals the product of two independent uniforms on $(0, 1)$ with probability $1 - p(Z)$, where:

\[p(z) = \frac{z(1 - z) A'(z)}{A(z) g_Z(z)}\]

Since $g_Z(z)$ is the derivative of the cumulative distribution function of $Z$, it holds that $0 \leq p(z) \leq 1$.

For the class of Extreme Value Copulas, We follow the methodology proposed by Ghoudi,1998. page 191. [35]. Here, is a detailed algorithm for sampling from bivariate Extreme Value Copulas:

Algotithm 1: Bivariate Extreme Value Copulas

(1) Simulate $U_1,U_2 \sim \mathcal{U}[0,1]$

(2) Simulate $Z \sim G_Z(z)$

(3) Select $W=U_1$ with probability $p(Z)$ and $W=U_1U_2$ with probability $1-p(Z)$

(4) Return $X=W^{Z/A(Z)}$ and $Y=W^{(1-Z)/A(Z)}$

Note that all functions present in the algorithm were previously defined in previous sections to ensure that the implemented methodology has a solid theoretical basis.

ExtremeValueCopula{P}

struct GalambosCopula{P} <: ExtremeValueCopula{P}
A(C::GalambosCopula, t::Real) = 1 - (t^(-C.θ) + (1 - t)^(-C.θ))^(-1/C.θ) # You can define your own Pickands representation
param = 2.5
C = GalambosCopula(param)

samples = rand(C,1000)   # sampling
cdf(C,samples)           # cdf
pdf(C,samples)           # pdf