17.1 Copulas

Definition 17.1  
A two-dimensional copula is a function $ C: \, [0,1]^2 \to [0,1]$ with the following properties:
  1. For every $ u \in [0,1]$

    $\displaystyle C(0,u) = C(u,0) = 0 \, .$ (17.1)

  2. For every $ u \in [0,1]$

    $\displaystyle C(u,1) = u$   and$\displaystyle \quad C(1,u) = u \, .$ (17.2)

  3. For every $ (u_1,u_2), (v_1,v_2) \in [0,1]\times [0,1]$ with $ u_1 \le v_1$ and $ u_2 \le v_2$:

    $\displaystyle C(v_1,v_2) - C(v_1,u_2) - C(u_1,v_2) + C(u_1,u_2) \ge 0 \, .$ (17.3)

A function that satisfies the first property is called grounded. The third property describes the analog of a non-decreasing, one-dimensional function. A function with this property is thus called 2-increasing. An example of a non-decreasing function is max$ (u,v)$. It is easy to see that for $ u_1=u_2=0$, $ v_1=v_2=1$ the third property is not satisfied; thus this function is not 2-increasing. The function $ (u,v) \to (2 u -1)(2 v -1)$ is, on the other hand, 2-increasing, yet decreasing for $ v\in (0,\frac{1}{2})$.

The notation ``copula'' becomes clear in the following theorem from Sklar (1959).

Theorem 17.1 (Sklar's theorem)  
Let $ H$ be a joint distribution function with marginal distributions $ F_1$ and $ F_2$. A copula $ C$ exists if:

$\displaystyle H(x_1,x_2) = C(F_1(x_1),F_2(x_2))$ (17.4)

$ C$ is unique when $ F_1$ and $ F_2$ are continuous. Conversely for a given copula $ C$ and marginal distributions $ F_1$ and $ F_2$ the function $ H$ defined by (16.4) is the joint distribution function.

A particular example is the product copula $ \Pi$: two random variables $ X_1$ and $ X_2$ are independent if and only if

$\displaystyle H(x_1,x_2) = F_1(x_1) \cdot F_2(x_2)$ (17.5)

The product copula $ C=\Pi$ is given by:

$\displaystyle \Pi(u_1,\cdots,u_p) = \prod_{j=1}^p u_p$ (17.6)

Another important example of a copula is the Gaussian or normal copula:

$\displaystyle C^{\rm Gauss}_{\rho}(u_1, u_2) \stackrel{\mathrm{def}}{=} \int_{-...
...1^{-1}(u_1)} \int_{- \infty}^{\Phi_2^{-1}(u_2)} f_\rho(r_1,r_2) d r_2 d r_1\; ,$ (17.7)

Here $ f_{\rho}$ denotes the bivariate normal density function with correlation $ \rho$ and $ \Phi_j,\; j=1,2$ represents the gaussian marginal distribution. In the case $ \rho=0$ we obtain:

$\displaystyle C^{\rm Gauss}_0(u_1, u_2)$ $\displaystyle =$ $\displaystyle \int_{- \infty}^{\Phi_1^{-1}(u_1)} f_1(r_1) d r_1 \;
\int_{- \infty}^{\Phi_2^{-1}(u_2)} f_2(r_2) d r_2 \;$  
  $\displaystyle =$ $\displaystyle u_1\, u_2$ (17.8)
  $\displaystyle =$ $\displaystyle \Pi (u_1,u_2)$   if$\displaystyle \quad \rho = 0 \; .$  

An important class of copulas is the Gumbel-Hougaard Family, Hutchinson and Lai (1990), Nelsen (1999). This class is parameterized by

$\displaystyle C_{\theta}(u_1, u_2) \stackrel{\mathrm{def}}{=}\exp \left\{ - \left[ (-\ln u_1)^{\theta} + (-\ln u_2)^{\theta} \right]^{1 / \theta} \right\} \; .$ (17.9)

For $ \theta = 1$ we obtain the product copula: $ C_1(u_1,u_2) =
\Pi(u_1,u_2) = u_1 \, u_2$. For $ \theta \to \infty$ we obtain the minimum copula:

$\displaystyle C_{\theta}(u_1,u_2) {\longrightarrow}
\min(u_1,u_2) \stackrel{\mathrm{def}}{=}M(u_1,u_2).$

$ M$ is also a copula which dominates every other copula $ C$:

$\displaystyle C(u_1,u_2) \le M(u_1,u_2).$

$ M$ is therefore referred to as the Fréchet-Hoeffding upper bound. The two-dimensional function $ W(u_1,u_2) \stackrel{\mathrm{def}}{=}
\max(u_1+u_2-1,0)$ satisfies:

$\displaystyle W(u_1,u_2) \le C(u_1,u_2)$

for all copulas. $ W$ is therefore called the Fréchet-Hoeffding lower bound.

Theorem 17.2  
Let $ C$ be a copula. Then for every $ u_1, u_2, v_1, v_2 \in [0,1]$ the following Lipschitz conditions hold:

$\displaystyle \vert C(u_2, v_2) - C(u_1,v_1)\vert \le \vert u_2 - u_1\vert + \vert v_2 - v_1\vert \, .$ (17.10)

Moreover the differentiability of the copulas can be shown.

Theorem 17.3  
Let $ C$ be a copula. For every $ u \in [0,1]$, the partial derivative $ \partial \, C/ \partial \, v$ exists almost everywhere in $ [0,1]$. In addition it holds that:

$\displaystyle 0 \le \frac{\partial}{\partial \, v} C(u,v) \le 1 \, .$ (17.11)

A similar statement for the partial derivative $ \partial \, C/ \partial \, u$ can be made. The function $ C_v(u) \stackrel{\mathrm{def}}{=}\partial \, C(u,v) / \partial \, v$ of $ u$ and $ C_u(v) \stackrel{\mathrm{def}}{=}\partial \, C(u,v) / \partial \, u$ of $ v$ are non-decreasing almost everywhere in [0,1].

To illustrate this theorem we consider the Gumbel-Hougaard copula (16.9):

$\displaystyle C_{\theta, u} (v)$ $\displaystyle =$ $\displaystyle \frac{\partial}{\partial \, u} C_{\theta}(u, v)
= \exp \left\{ - \left[ (-\ln u)^{\theta} + (-\ln v)^{\theta}
\right]^{1 / \theta} \right\} \times$  
    $\displaystyle \left[ (-\ln u)^{\theta} + (-\ln v)^{\theta}
\right]^{- \frac{\theta -1}{\theta}} \;
\frac{(- \ln u)^{\theta -1}}{u}.$ (17.12)

It is easy to see that $ C_{\theta,u}$ is a strictly monotone increasing function of $ v$. The inverse $ C_{\theta,u}^{-1}$ is therefore well defined. How do copulas behave under transformations? The next theorem gives some information on this.

Theorem 17.4  
Let $ X_1$ and $ X_2$ be random variables with continuous density functions and the copula $ C_{X_1 X_2}$. If $ T_1,T_2$ are strictly, monotone increasing transformations in the region of $ X_1$ and $ X_2$, then it holds that $ C_{T_1(X_1) \, T_2(X_2)} = C_{X_1 X_2}$. In other words: $ C_{X_1 X_2}$ is invariant under strictly, monotone increasing transformations of $ X_1$ and $ X_2$.