2.1 Definition

The Gumbel copula is defined as \[ C_\alpha\left(u,v\right)=\exp\left[-\left(\left(-\ln u\right)^\alpha+\left(-\ln v\right)^\alpha\right)^{\frac{1}{\alpha}}\right], \] where \(\alpha\geq1\) is the parameter.

2.2 Archimedean copula

The generator is \(\phi_\alpha(t)=\left(-\ln t\right)^\alpha\) with \(\alpha\geq1\) and \(t\in [0,1]\).

Thus, \(\phi\) is a continuous, strictly decreasing function from \([0,1]\) to \([0,+\infty]\), convex, and it is a strict generator.

We retrieve the Gumbel family through the relation \(\phi\alpha^{-1}\left(\phi\alpha(u)+\phi_\alpha(v)\right)\). Archimedean copulas form families of copulas that possess several interesting properties.

2.3 Density

We know that the density of an Archimedean copula, with a twice-differentiable generator is given by \[ c_\alpha(u,v)=-\frac{ \phi_{\alpha}^\prime\left(C_\alpha(u,v)\right) \phi_{\alpha}^\prime(u) \phi_{\alpha}^\prime(v)}{ \phi_{\alpha}^\prime\left(C_\alpha(u,v)\right)^3} \] This leads to the following density

The supra expression of the density is not valid on the edges of the square \([0,1] \times [0,1]\). On the edge of the domain, we have the following expressions \[ \forall u>0,v<1, c_\alpha(u,0)=c_\alpha(0,u) = c_\alpha(1,v) =c_\alpha(v,1)=0, \] but the density is infinite at \((0,0)\) and \((1,1)\).

2.4 Dependence measure

2.5 Extreme copula

The concept of tail dependence provides information about the “amount” of dependence at the distribution tails. It is a relevant tool for studying the simultaneous occurrence of extreme values. This is a local measure, unlike Kendall’s tau and Spearman’s rho, which measure dependence across the entire distribution. The coefficients of tail dependence, on the left and right, for a pair \((X,Y)\), are defined by \[ \lambda_L = \underset{t\rightarrow0^+}{\lim} P(Y > F_Y^{-1}(t) / X > F_X^{-1}(t)$ = 0 \text{and} \lambda_U = \underset{t\rightarrow1^-}{\lim} P(Y > F_Y^{-1}(t) / X > F_X^{-1}(t)$ = 2-2^{\frac{1}{\alpha}}. \]

The Gumbel copula is max-stable, i.e.  \[ C_\alpha\left(u^{\frac{1}{n}},v^{\frac{1}{n}}\right)^n = C_\alpha(u,v). \]

2.6 Multivariate copula

Using Nelsen (2006), the Gumbel copula is

2.7 Simulation

2.8 Frailty approach by Marshall and Olkin (1988)

Marshall’s approach involves using a random variable \(\Theta\) for a random vector \((X_1, \dots, X_d)\) such that the components \(X_i\) are conditionally independent given \(\Theta\). The joint distribution of the vector \((X_1, \dots, X_d)\) is given by \[ F_{X_1, \dots, X_d}(x_1, \dots, x_d)=L_\Theta\left( \sum_{i=1}^d L^{-1}_\Theta\left(F_{X_i}(x_i)\right) \right) \] where \(L(\Theta)\) denotes the Laplace transform of the random variable \(\Theta\). Moreover, the following algorithm allows for the simulation of random variables from an Archimedean copula \(L_\Theta^{-1}\):

• Simulate \((x_1, \dots, x_d) \stackrel{iid}{\sim} \text{Exp}(1)\),
• Simulate \(\theta\) following the distribution with Laplace transform \(\phi^{-1}\),
• For each \(i\), set \(u_i := \phi^{-1}\left(\frac{x_i}{\theta}\right)\).

In the case of the Gumbel copula, \(\phi^{-1}(t) = e^{-t^{1/\alpha}}\), which is the Laplace transform of a stable law with parameters \((1/\alpha, 0, 1, 0)\). For more details on stable laws, refer to Nolan (2005). Finally, to simulate random variables with a stable distribution, we used Chambers, Mallows, and Stuck (1976)’s algorithm.

3.1 Method of Moments - gumbel.MBE

This method involves estimating the parameters \(\theta\) of the marginal laws and the parameter \(\alpha\) of the copula using the method of moments, i.e.,

• solve \[ \left\{ \begin{array}{c} \overline X_n = f(\theta_1,\dots,\theta_d)\\ S_n^2 = g(\theta_1,\dots,\theta_d)\\ \mu_{3,n} = h(\theta_1,\dots,\theta_d)\\ \vdots\\ \end{array} \right. , \] where \(d\) is the length of \(\theta\).

repeat this step over all marginal and then - Invert Kendall’s tau or Spearman’s rho to obtain the parameter \(\alpha\) of the copula.

Example with exponential marginal, we get \[ \hat \lambda_n = \frac{1}{\overline X_n} \textrm{~~and~~} \hat \alpha_n = \frac{1}{1-\tau_n}. \]

3.2 Exact max_imum likelihood - gumbel.EML

When the density of the copula ex_ists, max_imum likelihood estimators can be used. To simplify, we assume that a bivariate copula with a density is used, and that the marginal distributions have densities. Let \(\theta_1\) and \(\theta_2\) denote the parameters of the marginal laws. The log-likelihood is written as: \[\begin{multline*} \ln \mathcal{L}(\alpha, \theta_1, \theta_2, x_1, \dots, x_n, y_1, \dots, y_n) = \sum_{i=1}^n \ln\left( c \left( F_1(x_i, \theta_1), F_2(y_i, \theta_2), \alpha \right) \right) \\ \sum_{i=1}^n \ln\left( f_1(x_i, \theta_1) \right) + \sum_{i=1}^n \ln\left( f_2(y_i, \theta_2) \right). \end{multline*}\] Often, explicit expressions for the estimators that max_imize \(\ln \mathcal{L}\) do not ex_ist, and a numerical max_imization is performed instead.

3.3 Inference for margins - gumbel.IFM

Under the assumption that the copula has a density, it is possible to combine the two previous approaches by first estimating the parameters of the marginal distributions, then estimating the copula parameter. This involves:

• Estimating the parameters \(\theta_1\) and \(\theta_2\) via max_imum likelihood
• Constructing pseudo-data \(\forall 1 \leq i \leq n,; u_i = F_1(x_i, \hat{\theta}_1)\) and \(v_i = F_2(y_i, \hat{\theta}_2)\)
• Estimating the parameter \(\alpha\) by maximizing the log-likelihood \[ \ln \mathcal{L}(\alpha, u1, \dots, u_n, v_1, \dots, v_n) = \sum_{i=1}^n \ln\left( c \left( u_i, v_i, \alpha \right) \right). \]

This method has the advantage of using the “classical” max_imum likelihood estimators for the marginals.

3.4 Canonical max_imum likelihood - gumbel.CML

It is a semi-parametric method based on the previous approach:

• Calculate the empirical distribution functions \(F_{1,n}\) and \(F_{2,n}\)
• Construct the pseudo-data \(\forall 1 \leq i \leq n,; u_i = F_{1,n}(x_i)\) and \(v_i = F_{2,n}(y_i)\)
• Estimate the parameter \(\alpha\) by maximizing the log-likelihood \[ \ln \mathcal{L}(\alpha, u1, \dots, u_n, v_1, \dots, v_n) = \sum_{i=1}^n \ln\left( c \left( u_i, v_i, \alpha \right) \right). \]