diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index c2589ea..0417d0b 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-27T15:54:21","documenter_version":"1.1.2"}} \ No newline at end of file +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-10-30T17:58:27","documenter_version":"1.1.2"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index b8c96d7..ab82165 100644 --- a/dev/index.html +++ b/dev/index.html @@ -25,4 +25,4 @@ author={Durante, Fabrizio and Sempi, Carlo}, year={2015}, publisher={CRC press} -}source
WilliamsonTransforms.𝒲 β€” Type
𝒲(X,d)(x)

Computes the Williamson d-transform of the random variable X, taken at point x.

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 an archimedean generator, able to generate archimedean copulas in dimensions up to $d$.

References:

  • Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581
  • McNeil, Alexander J., and Johanna NeΕ‘lehovΓ‘. "Multivariate Archimedean copulas, d-monotone functions and β„“ 1-norm symmetric distributions." (2009): 3059-3097.
source
WilliamsonTransforms.𝒲₋₁ β€” Type
𝒲₋₁(Ο•,d)

Computes the inverse Williamson d-transform of the d-monotone archimedean generator Ο•.

A $d$-monotone archimedean generator is a function $\phi$ on $\mathbb R_+$ that has these three properties:

  • $\phi(0) = 1$ 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.

For such a function $\phi$, the inverse Williamson-d-transform of $\phi$ is the cumulative distribution function $F$ of a non-negative random variable $X$, defined by :

\[F(x) = 𝒲_{d}^{-1}(\phi)(x) = 1 - \frac{(-x)^{d-1} \phi_+^{(d-1)}(x)}{k!} - \sum_{k=0}^{d-2} \frac{(-x)^k \phi^{(k)}(x)}{k!}\]

We return this cumulative distribution function in the form of the corresponding random variable <:Distributions.ContinuousUnivariateDistribution from Distributions.jl. You may then compute : - The cdf via Distributions.cdf - The pdf via Distributions.pdf and the logpdf via Distributions.logpdf - Samples from the distribution via rand(X,n)

References: - Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581 - McNeil, Alexander J., and Johanna NeΕ‘lehovΓ‘. "Multivariate Archimedean copulas, d-monotone functions and β„“ 1-norm symmetric distributions." (2009): 3059-3097.

source
+}source
WilliamsonTransforms.𝒲 β€” Type
𝒲(X,d)(x)

Computes the Williamson d-transform of the random variable X, taken at point x.

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 an archimedean generator, able to generate archimedean copulas in dimensions up to $d$.

References:

  • Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581
  • McNeil, Alexander J., and Johanna NeΕ‘lehovΓ‘. "Multivariate Archimedean copulas, d-monotone functions and β„“ 1-norm symmetric distributions." (2009): 3059-3097.
source
WilliamsonTransforms.𝒲₋₁ β€” Type
𝒲₋₁(Ο•,d)

Computes the inverse Williamson d-transform of the d-monotone archimedean generator Ο•.

A $d$-monotone archimedean generator is a function $\phi$ on $\mathbb R_+$ that has these three properties:

  • $\phi(0) = 1$ 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.

For such a function $\phi$, the inverse Williamson-d-transform of $\phi$ is the cumulative distribution function $F$ of a non-negative random variable $X$, defined by :

\[F(x) = 𝒲_{d}^{-1}(\phi)(x) = 1 - \frac{(-x)^{d-1} \phi_+^{(d-1)}(x)}{k!} - \sum_{k=0}^{d-2} \frac{(-x)^k \phi^{(k)}(x)}{k!}\]

We return this cumulative distribution function in the form of the corresponding random variable <:Distributions.ContinuousUnivariateDistribution from Distributions.jl. You may then compute : - The cdf via Distributions.cdf - The pdf via Distributions.pdf and the logpdf via Distributions.logpdf - Samples from the distribution via rand(X,n)

References: - Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581 - McNeil, Alexander J., and Johanna NeΕ‘lehovΓ‘. "Multivariate Archimedean copulas, d-monotone functions and β„“ 1-norm symmetric distributions." (2009): 3059-3097.

source