Copulae — Concepts and examples

Definitions and basic properties

Definition 2 — Copula density

Given a differentiable copula \(C\) the copula density \(c\) is defined by: \begin{equation} c(u_1, ..., u_d) = \frac{\partial^dC(u_1, ..., u_d)}{\partial u_1...\partial u_d}. \label{eq:copula_density} \end{equation}

It follows that \(C(u_1, ..., u_d)\) is monotonically increasing in each component \(u_i=C(1,...,u_i, ..., 1)\). Moreover, the copula function \(C\) is isotonic, that is \(u_j\le v_j\ \forall j \implies C(u_1, ..., u_d)\le C(v_1, ..., v_d)\). Finally, notice that (under mild regularity conditions) one has the following proposition.


In order to better understand the significance of Definition 1 let us consider a two-dimensional random vector \((X_1, X_2)\) with \(X_j\sim F_j\) and denoting the joint CDF by \(F\). Applying the probability integral transform one can find two marginals \(U_j=F_j(X_j)\) uniform on the unit interval so that one can write \begin{equation} F(x_1, x_2)=C(F_1(x_1), F_2(x_2)). \label{eq1} \end{equation} Therefore, a copula is the recipe needed to link the marginals to the joint distribution.

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Exercise 1

Show that \(U_j=F_j(X_j)\) implies \(U_j\sim\mathcal{U}[0,1]\).
Solution 1

Exercise 2

Show that equation \eqref{eq1} holds.
Solution 2

Exercise 3

Consider two uniform random variables \(U_1\) and \(U_2\) with known differentiable copula \(C\). Assuming \(U_1=u_1\) is observed, prove \eqref{eq:conditional_copula} in Proposition 3 for this \(d=2\) case.
Solution 3

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Equation \eqref{eq1} can be formalised into the following theorem.


One has the following corollaries.

Fundamental copulae

Let us now consider three fundamental examples corresponding to the cases of perfect dependence and independence. As we shall discuss, these are extreme cases which provide bounds to any other copula, and constitute therefore preliminary definitions for some results which will be presented later. First we consider the case of independent random variables.

Definition 5 — Independence copula

The joint distribution of \(d\) independent random variables is defined by the independence copula \begin{equation} C_{\text{ind}}(u_1, ..., u_d) = \prod_{i=1}^d u_i. \label{eq:independence_copula} \end{equation}

When all the random variables considered are deterministically related \(X_j = T_{ji}(X_i)\) for some set of strictly increasing transformations \(T_{ji}=T_{ij}^{-1}\), one can derive a relation between each pair of cumulative distribution functions as which for all \(i,j\) implies \[U_i = F_i(X_i) = F_j(T_{ji}(X_i)) = F_j(X_j) = U_j.\] Similarly, consider two random variables such that \(X_2=D(X_1)\); for any strictly decreasing transformation \(D\) one has which implies This motivates the definition of the following comonotonicity and counter-monotonicity copulae.

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Exercise 4

Show that given two independent random variables \(X_1\) and \(X_2\) the associated joint multivariate cumulative distribution function takes indeed the form \eqref{eq:independence_copula} from Definition 3.
Solution 4

Exercise 5

Show that Definition 6 indeed is the copula associated with perfectly correlated random variables \(U_1=U_2\) for the case \(d=2\).
Solution 5

Exercise 6

Show that \eqref{eq:counter_monotonicity_copula} in Definition 7 is indeed the copula associated with two random variables related by \(U_2=1-U_1\).
Solution 6

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As mentioned at the beginning of this section, these are extreme cases which are fundamental as they provide bounds to any other copula. This concept is formalised by the following theorem.

Theorem 8 — Fréchet-Hoeffding bounds

Given a copula \(C(\mathbf{u}) = C(u_1,..., u_d)\) one always has \begin{equation} W_{\text{counter}}(\mathbf{u})\le C(\mathbf{u}) \le C_{\text{co}}(\mathbf{u}). \label{eq:Frechet_Hoeffding_bounds} \end{equation}

Finally, let us consider the following proposition.

Proposition 9

Consider a random vector \(\mathbf{X}\) with copula \(C\), and let \(T_1, ..., T_d\) be a set of \(d\) strictly increasing transformations. Then, the copula of the transformed vector \((T_1(X_1), ..., T_d(X_d))\) is also \(C\).

Parametric families of copulae

The following table provides some examples of notable parametric families.


We can recognise two main classes of copulae: elliptical and archimedean. Elliptical copulae are implicit copulae arising from elliptical distributions (such as the normal, Student-t, and mixtures of normal distribution) via Sklar's theorem. Elliptical copulae have the advantage of allowing for flexible modelling of pairwise dependency, are simple to sample from, and usually densities are known explicitely. On the other hand, the copula itself is not explicit, and is characterised by radial symmetry, which implies identical lower and upper tail behaviour, as it we shall discuss.

Archimedean copulae instead are explicit copulae which can be written in terms of some monotone generator \(\psi:[0,\infty[\rightarrow [0,1]\) such that \(\psi(0)=1\), \(\psi(\infty)=0\). The associated archimedean copula is then written as \begin{equation} C(u_1, ..., u_d) = \psi\left(\sum_{i=1}^d\psi^{[-1]}(u_i)\right), \end{equation} with the pseudo-inverse \(\psi^{[-1]}(x)=\psi^{-1}(x)\mathbb{1}_{x\le\psi(0)}\). As an example, the generators of the archimedean copulae in the table above are: \begin{align*} \psi_{\text{Clayton}}(x) &= \theta^{-1}(x^{-\theta}-1),\newline \psi_{\text{Gumbel}}(x) &= (-\log x)^\theta,\newline \psi_{\text{Joe}}(x) &= -\log(1-(1-x)^\theta),\newline \psi_{\text{Frank}}(x) &= -\log\frac{e^{\theta x}-1}{e^{-\theta}-1}. \end{align*} The explicit nature of Archimedean copulae makes them particularly suitable for closed form calculations, allowing for many of their properties to be explicitely computed in term of the generator \(\psi\). Moreover they are not anymore restricted to radial symmetries.

Plot code

import numpy as np, matplotlib.pyplot as plt, seaborn as sns
import scipy.stats as st
sns.set()

n = 1000
x1 = x2 = np.linspace(-2.7, 2.7, n)
dx = x1[-1] - x2[-2]
u1, u2 = st.norm.cdf(x1), st.norm.cdf(x2)

# INDEPENDENCE COPULA
def Ind_cop(u1, u2):
    U1, U2 = np.meshgrid(u1, u2)
    return U1 * U2

# GAUSSIAN COPULA
def Gauss_cop(u1, u2, rho=0.5):
    """Rho in [0, 1]"""
    iU1, iU2 = np.meshgrid(st.norm.ppf(u1), st.norm.ppf(u2))
    jnt = st.multivariate_normal.cdf(np.array([iU1, iU2]).T,
                                     mean=[0, 0], cov=[[1, rho], [rho, 1]])
    return jnt

# CLAYTON COPULA
def Clayton_cop(u1, u2, theta=1.):
    """Theta in [-1, inf[ \ {0}"""
    U1, U2 = np.meshgrid(u1, u2)
    jnt = (U1**(-theta) + U2**(-theta) - 1).clip(min=0) ** (-1/theta)
    return jnt

# GUMBEL COPULA
def Gumbel_cop(u1, u2, theta=1.):
    """Theta in [1, inf["""
    U1, U2 = np.meshgrid(u1, u2)
    jnt = np.exp(-((-np.log(U1))**theta
                   +(-np.log(U2))**theta)**(1/theta))
    return jnt

# JOE COPULA
def Joe_cop(u1, u2, theta=1.):
    """Theta in [1, inf["""
    U1, U2 = np.meshgrid(u1, u2)
    jnt = 1 - ((1-U1)**theta + (1-U2)**theta - ((1-U1)*(1-U2))**theta) ** (1/theta)
    return jnt

# FRANK COPULA
def Frank_cop(u1, u2, theta=1.):
    """Theta in [-inf, inf[ \ {0}"""
    U1, U2 = np.meshgrid(u1, u2)
    jnt = -(1/theta) * np.log(1+((np.exp(-theta*U1) - 1)*(np.exp(-theta*U2) - 1))/(np.exp(-theta) - 1))
    return jnt

J = Clayton_cop(u1, u2, theta=5)
dJ = (np.diff(np.diff(J).T).T).clip(min=1e-10)
plt.contour(x1[1:], x2[1:], dJ, 10)
plt.show()

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Exercise 7

Given a vector \((X_1, X_2)\) from the standard bivariate normal distribution with correlation \(\rho\), derive the gaussian copula presented in the table above.
Solution 7

Exercise 8

Show that the limiting copula of the Clayton copula \(C_\theta^{\text{Clayton}}(u_1, u_2)\) for \(\theta\rightarrow\infty\) is the comonotonicity copula \(C_{\text{co}}\) and for \(\theta\rightarrow -1\) is the countermonotonicity copula \(W_{\text{counter}}\). Moreover show that for \(\theta\rightarrow 0\) Clayton's copula tends to the independence copula.
Solution 8

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Simulating from Copulae

Let us now consider the essential recipes for drawing uniform random vectors \((U_1, ..., U_d)\) with a desired copula and uniform marginals, which can be useful e.g. to perform Monte Carlo simulations. Clearly, arbitrary marginals can be obtained via the quantile transformation as dictated by the probability integral transform discussed earlier.

Implicit copulae

For implicit copulae derived from a multivariate distribution such as the Gaussian and Student-t the procedure is quite straightforward.

Algorithm 10 — Gaussian Copula

1. Given a correlation matrix \(\Pi\) perform a Cholesky decomposition, that is, find \(\Lambda\) such that \(\ \Pi = \Lambda^T\Lambda\);
2. Generate iid standard normals \(\mathbf{G}= (G_1, ..., G_d)^T\);
3. Compute \(\ \mathbf{X} = \Lambda\mathbf{G}\);
4. Return the desired vector of \(\ U_i = \Phi(X_i)\), with \(\Phi\) the standard normal cumulative distribution function.

Algorithm 11 — Student-t Copula

1. Given a correlation matrix \(\ \Pi\), generate a normal vector \(\mathbf{X}\) performing steps (1) to (4) from Algorithm 10;
2. Generate \(\xi=\sum_{i=1}^\nu Y_i^2\), with \(Y_i \stackrel{\text{iid}}{\sim} \mathcal{N}(0,1)\), so that \(\xi\stackrel{\text{iid}}{\sim}\chi_\nu^2\);
3. Return the desired vector \(\ U_i = t_\nu(X_i/\sqrt{\xi/\nu})\), with \(t_\nu\) the univariate Student-t cumulative distribution function.

Explicit copulae

The Archimedean copulae we have considered are actually part of a class of copulae known as Laplace Transform Archimedean Copulae (LT-Archimedean for short) where the inverse of the generator \(\phi^{[-1]}\) can be represented as a Laplace transform of some function \(G\): \begin{equation} \phi^{[-1]}(x) = \int_0^\infty e^{-tx}\text{d}G(t). \label{eq:LT-A} \end{equation} More specifically, one can show [4] that all completely monotone Archimedean generators are Laplace-Stieltjes transforms of distribution functions on \(\mathbb{R}^+\). With this in mind, we are now ready to consider an algorithm to sample from such copulae.

Algorithm 12 — LT-Archimedean Copulae

Given an LT-Archimedean copula with generator \(\phi\),

1. Find a CDF \(G\) which satisfies \eqref{eq:LT-A};
2. Generate a variate \(V\sim G\);
3. Generate uniform iid \(X_1, ..., X_d\);
4. Return the vector with elements \(U_i = \phi^{[-1]}\left(-\frac{\log X_i}{V}\right)\).

See [1] and [5] for more details.

For the special case of \(d=2\) one has however a simpler approach.

Algorithm 13 — 2D Explicit Copulae

Consider the conditional distribution \(F_{1}=\mathbb{P}(U_1\le u_1 | U_2=u_2)\). Recalling \eqref{eq:conditional_copula} from Proposition 3, this is equivalent to \begin{equation*} F_{1} = c_{1}(u_1, u_2) = \frac{\partial}{\partial u_2} C(u_1, u_2). \end{equation*} Therefore one obtains the desired \((U_1,\ U_2)\) via the following algorithm:

1. Sample \(u_1, t \overset{\text{iid}}{\sim} \mathcal{U}[0,1]\);
2. Compute \(u_2 = c_{1}^{[-1]}(u_1, t)\);

Example 1: gaussian copula and gaussian marginals

The simplest example is that in which we sample from a gaussian joint distribution, since in most programming languages this is often well implemented in libraries. Fig.3 below presents an example where the marginals follow standard normal distributions and the gaussian copula has linear correlation \(\rho=0.75\).

Plot code

import pandas as pd, matplotlib.pyplot as plt, seaborn as sns
import scipy.stats as st
N = 1_000_000
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = pd.DataFrame(mvnorm.rvs(N), columns=['X1', 'X2'])

jnt_kw = dict(height=5, ratio=5, space=0,
              color="tab:blue", alpha=0)

h = (sns.jointplot(data=X, x='X1', y='X2', marginal_kws=dict(bins=30),
                   xlim=(-3,3), ylim=(-3,3), kind='hex', gridsize=2, **jnt_kw)
        .plot_joint(sns.kdeplot, shade=False, cmap='viridis'))
plt.show()

Example 2: gaussian copula and non-gaussian marginals

In order to produce samples from a joint with gaussian copula but non-gaussian marginals we first need to obtaint the copula's uniform marginals \(U_i = \Phi(X_i)\).

Plot code

import pandas as pd, matplotlib.pyplot as plt, seaborn as sns
import scipy.stats as st
N = 1_000_000
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = pd.DataFrame(mvnorm.rvs(N), columns=['X1', 'X2'])

jnt_kw = dict(height=5, ratio=5, space=0,
              color="tab:blue", alpha=0)

U = pd.DataFrame(st.norm().cdf(X), columns=['U2', 'U1'])

h = (sns.jointplot(data=U, x='U1', y='U2', marginal_kws=dict(bins=30),
                   xlim=(0,1), ylim=(0,1),kind='hex', gridsize=2, **jnt_kw)
        .plot_joint(plt.hexbin, gridsize=30, cmap='Blues'))
plt.show()

Finally, the copula's uniforms can be transformed to the desired marginals. Take as an example \(\widetilde{X}_1\sim \text{Beta}(\alpha=3, \beta=10)\), and \(\widetilde{X}_2\sim \text{Gumbel}(\mu=0, \beta=1)\): Figure 5 below plots the joint density with these marginals and gaussian copula with \(\rho=0.75\), while Figure 6, plots these two marginals on the independence copula.

Plot code

import pandas as pd, matplotlib.pyplot as plt, seaborn as sns
import scipy.stats as st
N = 1_000_000
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = pd.DataFrame(mvnorm.rvs(N), columns=['X1', 'X2'])

jnt_kw = dict(height=5, ratio=5, space=0,
              color="tab:blue", alpha=0)

U = pd.DataFrame(st.norm().cdf(X), columns=['U2', 'U1'])

# With Gaussian Copula
x1_t = st.beta(a=3, b=10).ppf(U.U1)
x2_t = st.gumbel_l().ppf(U.U2)
X_trans = pd.DataFrame([x1_t, x2_t], index=['XT1', 'XT2']).T

h = (sns.jointplot(data=X_trans, x='XT1', y='XT2', kind='hex', gridsize=2,
                   marginal_kws=dict(bins=30),
                   xlim=(0, 0.6), ylim=(-5, 2.0),
                   **jnt_kw)
        .plot_joint(sns.kdeplot, cmap='viridis'))
plt.show()

# With Independence Copula
x1, x2 = st.beta(a=3, b=10).rvs(N), st.gumbel_l().rvs(N)
h = (sns.jointplot(x=x1, y=x2, kind='hex', marginal_kws=dict(bins=30),
                   xlim=(0,0.6), ylim=(-5,2.0),
                   **jnt_kw)
        .plot_joint(sns.kdeplot, cmap='viridis'))
plt.show()

Example 3: non-gaussian copula and non-gaussian marginals

At last let's replicate the last exercise replacing the gaussian copula with e.g. a Clayton copula. One can find [5] the correct cumulative distribution function \(G\) to be considered in this case is that of a Gamma distribution \(\Gamma(\theta^{-1},1)\).

Plot code

import pandas as pd, matplotlib.pyplot as plt, seaborn as sns
import scipy.stats as st

jnt_kw = dict(height=5, ratio=5, space=0,
              color="tab:blue", alpha=0)

N = 1_000_000
θ = 2
def clayton(θ, N, d=2):
    Γ = st.gamma.rvs(a=1/θ, scale=1, size=N)
    X = st.uniform.rvs(size=(d,N))
    return (1 - np.log(X) / Γ) ** (-1/θ)

U1, U2 = clayton(θ=θ, N=N, d=2)
U = pd.DataFrame([U1, U2], index=['U1', 'U2']).T

h = (sns.jointplot(data=U, x='U1', y='U2', marginal_kws=dict(bins=30),
                   xlim=(0,1), ylim=(0,1),
                   kind='hex', gridsize=2, **jnt_kw)
        .plot_joint(plt.hexbin, gridsize=30, cmap=my_cmap))
plt.show()


x1_t = st.beta(a=3, b=10).ppf(U.U1)
x2_t = st.gumbel_l().ppf(U.U2)
X_trans = pd.DataFrame([x1_t, x2_t], index=['XT1', 'XT2']).T

h = (sns.jointplot(data=X_trans, x='XT1', y='XT2', kind='hex', gridsize=2,
                   marginal_kws=dict(bins=30),
                   xlim=(0, 0.6), ylim=(-6, 2.0),
                   ** jnt_kw)
        .plot_joint(sns.kdeplot, cmap='viridis'))
plt.show()

In Python, the package OpenTURNS provides a fast implementation of copulas. As an example, one can perform sampling from the Clayton copula we have considered so far simply as it follows:

import openturns as ot, numpy as np

X = ot.ComposedDistribution([ot.Uniform(0, 1),
                             ot.Uniform(0, 1)],
                            ot.ClaytonCopula(2))

U1, U2 = np.array(X.getSample(1_000_000)).T

As usual the two uniform marginals can now be transformed into whichever marginals one might desire.

Estimation: Fitting Copulae to Data

The task of fitting a copula to data is a rather challenging one. Generally speaking, one approach could follow a Maximum Likelihood Estimation principle. For example, given the random vector \(\mathbf{X}=(X_1, ..., X_d)\), and assuming parametric models for the copula density \(c_{\mathbf{X}}(\cdot|\mathbf{\theta}_C)\) and for the marginal cumulative distribution functions \(F_1(\cdot|\mathbf{\theta}_1), ..., F_d(\cdot|\mathbf{\theta}_d)\), one can write an expression for the density of \(\mathbf{X}\) as in \eqref{eq:density_from_sklar}: \[ f_\mathbf{X}(\mathbf{x})=c_{\mathbf{X}}(F_{X_1}(x_1), ..., F_{X_d}(x_d))\prod_{i=1}^d f_{X_i}(x_i). \] Given a vector of \(n\) observations \(\mathbf{D}_{1:n}=(\mathbf{D}_1, ..., \mathbf{D}_n)\) one can then attempt to maximise the log-likelihood so as to obtain the ML estimators \(\hat{\mathbf{\theta}}_1, ..., \hat{\mathbf{\theta}}_d, \hat{\mathbf{\theta}}_C\). The problem with this approach is found in the very large number of parameters to be estimated, which complicates optimisation. Moreover, the mispecification of one single univariate distributions \(F_{X_i}(\cdot|\mathbf{\theta}_i)\) can introduce biases for all other marginals as well as for the copula.
One alternative is provided by pseudo-MLE whereby one splitts the estimation into a two-step procedure, estimating the marginals first, and the copula in a second stage. In particular, one starts by estimating the marginal cumulative distribution functions \(F_{X_i}(\cdot|\mathbf{\theta}_i)\) yielding the estimator \(\hat{F}_{X_i}\), either using a parametric model to obtain a univariate ML-estimator for \(\hat{\mathbf{\theta}}_i\), or even considering the empirical cumulative distribution function. Then, one estimates the copula as a second step, again maximising the log-likelihood
Yet another alternative is moment-matching.

Proposition 14

For Gaussian and Student-t distributed \(\mathbf{X}=(X_1, ..., X_d)\), and Gaussian or Student-t copulae with covariance matrix \(\Omega\), the following holds: \begin{equation} \rho_\tau(X_i, X_j) = \frac{2}{\pi}\arcsin(\Omega_{ij}), \end{equation} where \(\rho_\tau\) denotes Kendall's tau.

A number of non-parametric methods exist as well, such as the empirical Bernstein copula [6] as well as kernel smoothing estimators. The Python package OpenTURNS provides utilities to fit these non-parametric copula models.

Estimation in practice: parametric estimates

Practically, python has a number of packages (although still in their infancy) which implement similar methods. Popular ones are Copulas, Copulae and OpenTURNS. Let's consider a few example with OpenTURNS. The simplest case is that of estimating the gaussian copula associated with a multivariate normal distribution, which effectively boils down to estimating the correlation matrix. The code

import openturns as ot, numpy as np, scipy.stats as st

np.random.seed(seed=0)
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = mvnorm.rvs(10_000)

dist = ot.NormalCopulaFactory().build(X)
print(dist)

returns The parameters can then be extracted as

print(dist.getParameter())

which returns

      [0.754492]

Similarly, for data with gaussian copula but non-gaussian marginals:

import openturns as ot, numpy as np, scipy.stats as st

np.random.seed(seed=0)
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = mvnorm.rvs(10_000)
U = st.norm().cdf(X)
X_trans = np.array([st.beta(a=3, b=10).ppf(U[:,0]),
                    st.gumbel_l().ppf(U[:,1])]).T

dist = ot.NormalCopulaFactory().build(X_trans)
print(dist)

one obtains Notice that having different non-gaussian marginals does not affect the estimates of the parameters.

Finally, consider the following example for the non-gaussian Clayton copula with \(\theta=2\) and \(\text{Beta}(\alpha=3, \beta=10)\) and \(\text{Gumbel}(\mu=0,\beta=1)\) marginals: the code

import openturns as ot, numpy as np, scipy.stats as st

J = ot.ComposedDistribution([ot.Uniform(0, 1),
                             ot.Uniform(0, 1)],
                            ot.ClaytonCopula(2))
U1, U2 = np.array(J.getSample(10_000)).T

X_trans = np.array([st.beta(a=3, b=10).ppf(U1),
                    st.gumbel_l().ppf(U2)]).T

dist = ot.ClaytonCopulaFactory().build(X_trans)
print(dist)

returns

      ClaytonCopula(theta = 1.98901)
    

Estimation in practice: non-parametric estimates

As briefly mentioned OpenTURNS also allows us to estimate copulae non-parametrically. Let us start by sampling from a bivariate normal distribution with linear correlation coefficient \(\rho=0.75\):

import openturns as ot, numpy as np, scipy.stats as st

N = 1_000
mvnorm = st.multivariate_normal(mean=[0, 0], cov=[[1., 0.75],
                                                  [0.75, 1.]])
X = mvnorm.rvs(N)

Now, one can extract a Bernstein non-parametric estimate simply as

Bs_Est = ot.EmpiricalBernsteinCopula(X, 20, False)

where the second argument is the number of bins which each dimension of the unit cube \([0,1]^d\) is divided into in order to estimate the empirical copula. Similarly, one can obtain a Kernel Smoothing estimation:

KS_Est = ot.KernelSmoothing().build(X).getCopula()

It is then possible to obtain a new sample of N points from the estimated copulae Bs_Est and KS_Est as Bs_Est.getSample(N) and KS_Est.getSample(N) respectively.

References

[1] "Quantitative Risk Management: Concepts, Techniques and Tools", Alexander J. McNeil, Paul Embrechts, and Rüdiger Frey, 2005
[2] "An Introduction to Copulas", Roger B. Nelsen, 2006
[3] "Correlation and Dependence in Risk Management: Properties and Pitfalls", Paul Embrechts, Alexander McNeil, and Daniel Straumann, 1999
[4] "An Introduction to Probability Theory and Its Applications", W. Feller, 1971, volume 2, 2nd edition; John Wiley & Sons; p. 439
[5] “Families of Multivariate Distributions.”, A.W. Marshall, I. Olkin, 1988, Journal of the American Statistical Association, 83, 834–841
[6] “The Bernstein copula and its applications to modeling and approximations of multivariate distributions.”, Sancetta, A. and S. Satchell, 2004, Econometric Theory 20(3), 535–562

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