Archimedean Generators

WilliamsonGenerator{TX}
i𝒲{TX}

Fields:

• X::TX – a random variable that represents its Williamson d-transform
• d::Int – the dimension of the transformation.

Constructor

WilliamsonGenerator(X::Distributions.UnivariateDistribution, d)
i𝒲(X::Distributions.UnivariateDistribution,d)

The WilliamsonGenerator (alias i𝒲) allows to construct a d-monotonous archimedean generator from a positive random variable X::Distributions.UnivariateDistribution. The transformation, which is called the inverse Williamson transformation, is implemented in WilliamsonTransforms.jl.

For a univariate non-negative random variable $X$, with cumulative distribution function $F$ and an integer $d\ge 2$, the Williamson-d-transform of $X$ is the real function supported on $[0,\infty[$ given by:

\[\phi(t) = 𝒲_{d}(X)(t) = \int_{t}^{\infty} \left(1 - \frac{t}{x}\right)^{d-1} dF(x) = \mathbb E\left( (1 - \frac{t}{X})^{d-1}_+\right) \mathbb 1_{t > 0} + \left(1 - F(0)\right)\mathbb 1_{t <0}\]

This function has several properties:

• We have that $\phi(0) = 1$ and $\phi(Inf) = 0$
• $\phi$ is $d-2$ times derivable, and the signs of its derivatives alternates : $\forall k \in 0,...,d-2, (-1)^k \phi^{(k)} \ge 0$.
• $\phi^{(d-2)}$ is convex.

These properties makes this function what is called a d-monotone archimedean generator, able to generate archimedean copulas in dimensions up to $d$. Our implementation provides this through the Generator interface: the function $\phi$ can be accessed by

G = WilliamsonGenerator(X, d)
ϕ(G,t)

Note that you'll always have:

max_monotony(WilliamsonGenerator(X,d)) === d

References:

• [21] Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581
• [20] McNeil, Alexander J., and Johanna Nešlehová. "Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions." (2009): 3059-3097.

IndependentGenerator

Constructor

IndependentGenerator()
IndependentCopula(d)

The Independent Copula in dimension $d$ is the simplest copula, that has the form :

\[C(\mathbf{x}) = \prod_{i=1}^d x_i.\]

It happends to be an Archimedean Copula, with generator :

\[\phi(t) = \exp{-t}\]

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

MGenerator

Constructor

MGenerator()
MCopula(d)

The Upper Frechet-Hoeffding bound is the copula with the greatest value among all copulas. It correspond to comonotone random vectors.

For any copula $C$, if $W$ and $M$ are (respectively) the lower and uppder Frechet-Hoeffding bounds, we have that for all $\mathbf{u} \in [0,1]^d$,

\[W(\mathbf{u}) \le C(\mathbf{u}) \le M(\mathbf{u})\]

The two Frechet-Hoeffding bounds are also Archimedean copulas.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

WGenerator

Constructor

WGenerator()
WCopula(d)

The Lower Frechet-Hoeffding bound is the copula with the lowest value among all copulas. Note that $W$ is only a proper copula when $d=2$, in greater dimensions it is still the (pointwise) lower bound, but not a copula anymore. For any copula $C$, if $W$ and $M$ are (respectively) the lower and uppder Frechet-Hoeffding bounds, we have that for all $\mathbf{u} \in [0,1]^d$,

\[W(\mathbf{u}) \le C(\mathbf{u}) \le M(\mathbf{u})\]

The two Frechet-Hoeffding bounds are also Archimedean copulas.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

ClaytonGenerator{T}

Fields:

• θ::Real - parameter

Constructor

ClaytonGenerator(θ)
ClaytonCopula(d,θ)

The Clayton copula in dimension $d$ is parameterized by $\theta \in [-1/(d-1),\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = \left(1+\mathrm{sign}(\theta)*t\right)^{-1\frac{1}{\theta}}\]

It has a few special cases:

• When θ = -1/(d-1), it is the WCopula (Lower Frechet-Hoeffding bound)
• When θ = 0, it is the IndependentCopula
• When θ = ∞, is is the MCopula (Upper Frechet-Hoeffding bound)

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

FrankGenerator{T}

Fields:

• θ::Real - parameter

Constructor

FrankGenerator(θ)
FrankCopula(d,θ)

The Frank copula in dimension $d$ is parameterized by $\theta \in [-\infty,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = -\frac{\log\left(1+e^{-t}(e^{-\theta-1})\right)}{ heta}\]

It has a few special cases:

• When θ = -∞, it is the WCopula (Lower Frechet-Hoeffding bound)
• When θ = 1, it is the IndependentCopula
• When θ = ∞, is is the MCopula (Upper Frechet-Hoeffding bound)

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

GumbelGenerator{T}

Fields:

• θ::Real - parameter

Constructor

GumbelGenerator(θ)
GumbelCopula(d,θ)

The Gumbel copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = \exp{-t^{\frac{1}{θ}}}\]

It has a few special cases:

• When θ = 1, it is the IndependentCopula
• When θ = ∞, is is the MCopula (Upper Frechet-Hoeffding bound)

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

AMHGenerator{T}

Fields:

• θ::Real - parameter

Constructor

AMHGenerator(θ)
AMHCopula(d,θ)

The AMH copula in dimension $d$ is parameterized by $\theta \in [-1,1)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \frac{1-\theta}{e^{-t}-\theta}\]

It has a few special cases:

• When θ = 0, it is the IndependentCopula

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

JoeGenerator{T}

Fields:

• θ::Real - parameter

Constructor

JoeGenerator(θ)
JoeCopula(d,θ)

The Joe copula in dimension $d$ is parameterized by $\theta \in [1,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = 1 - \left(1 - e^{-t}\right)^{\frac{1}{\theta}}\]

It has a few special cases:

• When θ = 1, it is the IndependentCopula
• When θ = ∞, is is the MCopula (Upper Frechet-Hoeffding bound)

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

GumbelBarnettGenerator{T}

Fields:

• θ::Real - parameter

Constructor

GumbelBarnettGenerator(θ)
GumbelBarnettCopula(d,θ)

The Gumbel-Barnett copula is an archimdean copula with generator:

\[\phi(t) = \exp{θ^{-1}(1-e^{t})}, 0 \leq \theta \leq 1.\]

It has a few special cases:

• When θ = 0, it is the IndependentCopula

References:

• [4] Joe, H. (2014). Dependence modeling with copulas. CRC press, Page.437
• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

InvGaussianGenerator{T}

Fields:

• θ::Real - parameter

Constructor

InvGaussianGenerator(θ)
InvGaussianCopula(d,θ)

The Inverse Gaussian copula in dimension $d$ is parameterized by $\theta \in [0,\infty)$. It is an Archimedean copula with generator :

\[\phi(t) = \exp{\frac{1-\sqrt{1+2θ^{2}t}}{θ}}.\]

More details about Inverse Gaussian Archimedean copula are found in :

Mai, Jan-Frederik, and Matthias Scherer. Simulating copulas: stochastic models, sampling algorithms, and applications. Vol. 6. # N/A, 2017. Page 74.

It has a few special cases:

• When θ = 0, it is the IndependentCopula

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.