Archimedean Generators
WilliamsonGenerator
Copulas.WilliamsonGenerator
— TypeWilliamsonGenerator{TX}
i𝒲{TX}
Fields:
X::TX
– a random variable that represents its Williamson d-transformd::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:
IndependentGenerator
Copulas.IndependentGenerator
— TypeIndependentGenerator
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
Copulas.MGenerator
— TypeMGenerator
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
Copulas.WGenerator
— TypeWGenerator
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
Copulas.ClaytonGenerator
— TypeClaytonGenerator{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
Copulas.FrankGenerator
— TypeFrankGenerator{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
Copulas.GumbelGenerator
— TypeGumbelGenerator{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
Copulas.AMHGenerator
— TypeAMHGenerator{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
Copulas.JoeGenerator
— TypeJoeGenerator{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
Copulas.GumbelBarnettGenerator
— TypeGumbelBarnettGenerator{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:
InvGaussianGenerator
Copulas.InvGaussianGenerator
— TypeInvGaussianGenerator{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.
- [3]
- R. B. Nelsen. An Introduction to Copulas. 2nd ed Edition, Springer Series in Statistics (Springer, New York, 2006).
- [4]
- H. Joe. Dependence Modeling with Copulas (CRC press, 2014).
- [20]
- A. J. McNeil and J. Nešlehová. Multivariate Archimedean Copulas, d -Monotone Functions and L1 -Norm Symmetric Distributions. The Annals of Statistics 37, 3059–3097 (2009).
- [21]
- R. E. Williamson. On multiply monotone functions and their laplace transforms (Mathematics Division, Office of Scientific Research, US Air Force, 1955).