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. An extreme value copula $C$ has the property that $C(u_1^t, u_2^t)=(C(u_1,u_2))^t, t > 0,$ and can be represented by

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

or

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

Here, $\ell(\cdot)$ is the stable tail dependence function and $A$ is the Pickands dependence function where $A: \[0,1\] \to \[1/2, 1\]$ is a convex function satisfying $\max(t, t-1)\leq A(t) \leq 1.$ 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 TwinCopulas, 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 [the list of implemented extreme value copulas] 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 the following method is required:

• 𝘈(C::ExtremeValueCopula) =

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 ([Ghoudi1998]): 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) in the case where $A$ admits a second derivative.

Simulation of Bivariate Extreme Value Distributions

To simulate a bivariate extreme value distribution $C(x, y)$, 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 allows simulating such a distribution.

Assume $A$ has a second derivative, making the distribution $F$ 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 propose two different algorithms to generate samples from a copula.

Conditional Sampling for Bivariate Extreme Value Copulas

When this is defined, TwinCopulas uses Algorithm 1, introduced in [reference], to sample from the copula and its derivative as follows. We adapt Algorithm 1 from [reference] for the case of bivariate Extreme Value Copulas. The input for the algorithm is a bivariate Extreme Value Copula $C: [0,1]^2 \to [0,1].$

Algotithm 1: Bivariate Extreme Value Copulas

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

(2) Compute (the right continuous version of) the function $F_{U_1|U_2}(u_1)=\frac{\partial}{\partial u_2}C(u_1,u_2)=\frac{C(u_1,u_2)}{u_2}\left[A\left(\frac{\log(u_1)}{\log(u_1u_2)}\right) - \log(u_1)A'\left(\frac{\log(u_1)}{\log(u_1u_2)}\right)\right], \quad u_2 \in [0,1].$ (3) Compute de generalized inverse of $F_{U_1|U_2},$ i.e $F^{-1}_{U_1|U_2}(v)=\inf\{u_1 > 0: F_{U_1|U_2}(u_1)\geq v\}, \quad v \in [0,1].$ (4) Simulate $V \sim \mathcal{U}[0,1],$ independent of $U_2.$

(5) Set $U_1 = F^{-1}_{U_1|U_2}(V)$ and return $(U_1, U_2).$

Sampling from Archimedean Copula Using Frailty Distribution

Algorithm

Here, is a detailed algorithm for sampling from bivariate Extreme Value Copulas proposed by Ghoudi:

Algotithm 2: 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 $Y=W^{(1-Z)/A(Z)}$

We can use either of the two algorithms to generate random samples, and more specifically, by default, TwinCopulas uses Algorithm 2 to obtain samples from a bivariate extreme value copula.