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-{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-02-12T22:34:31","documenter_version":"1.2.1"}}
+{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-02-12T22:40:22","documenter_version":"1.2.1"}}

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@@ -25,4 +25,4 @@
   author={Durante, Fabrizio and Sempi, Carlo},
   year={2015},
   publisher={CRC press}
-}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/ffb98220fedf75a3b1170b7bcf7916bc8ec07890/src/WilliamsonTransforms.jl#L3-L47">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="WilliamsonTransforms.𝒲" href="#WilliamsonTransforms.𝒲"><code>WilliamsonTransforms.𝒲</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">𝒲(X,d)(x)</code></pre><p>Computes the Williamson d-transform of the random variable X, taken at point x.</p><p>For a univariate non-negative random variable <span>$X$</span>, with cumulative distribution function <span>$F$</span> and an integer <span>$d\ge 2$</span>, the Williamson-d-transform of <span>$X$</span> is the real function supported on <span>$[0,\infty[$</span> given by:</p><p class="math-container">\[\phi(t) = 𝒲_{d}(X)(t) = \int_{t}^{\infty} \left(1 - \frac{t}{x}\right)^{d-1} dF(x) = \mathbb E\left( (1 - \frac{t}{X})^{d-1}_+\right) \mathbb 1_{t &gt; 0} + \left(1 - F(0)\right)\mathbb 1_{t &lt;0}\]</p><p>This function has several properties:      - We have that <span>$\phi(0) = 1$</span> and <span>$\phi(Inf) = 0$</span>     - <span>$\phi$</span> is <span>$d-2$</span> times derivable, and the signs of its derivatives alternates : <span>$\forall k \in 0,...,d-2, (-1)^k \phi^{(k)} \ge 0$</span>.     - <span>$\phi^{(d-2)}$</span> is convex.</p><p>These properties makes this function what is called an <em>archimedean generator</em>, able to generate <em>archimedean copulas</em> in dimensions up to <span>$d$</span>. </p><p>References: </p><ul><li>Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581</li><li>McNeil, Alexander J., and Johanna Nešlehová. &quot;Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions.&quot; (2009): 3059-3097.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/ffb98220fedf75a3b1170b7bcf7916bc8ec07890/src/WilliamsonTransforms.jl#L17-L38">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="WilliamsonTransforms.𝒲₋₁" href="#WilliamsonTransforms.𝒲₋₁"><code>WilliamsonTransforms.𝒲₋₁</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">𝒲₋₁(ϕ,d)</code></pre><p>Computes the inverse Williamson d-transform of the d-monotone archimedean generator ϕ. </p><p>A <span>$d$</span>-monotone archimedean generator is a function <span>$\phi$</span> on <span>$\mathbb R_+$</span> that has these three properties:</p><ul><li><span>$\phi(0) = 1$</span> and <span>$\phi(Inf) = 0$</span></li><li><span>$\phi$</span> is <span>$d-2$</span> times derivable, and the signs of its derivatives alternates : <span>$\forall k \in 0,...,d-2, (-1)^k \phi^{(k)} \ge 0$</span>.</li><li><span>$\phi^{(d-2)}$</span> is convex.</li></ul><p>For such a function <span>$\phi$</span>, the inverse Williamson-d-transform of <span>$\phi$</span> is the cumulative distribution function <span>$F$</span> of a non-negative random variable <span>$X$</span>, defined by : </p><p class="math-container">\[F(x) = 𝒲_{d}^{-1}(\phi)(x) = 1 - \frac{(-x)^{d-1} \phi_+^{(d-1)}(x)}{k!} - \sum_{k=0}^{d-2} \frac{(-x)^k \phi^{(k)}(x)}{k!}\]</p><p>We return this cumulative distribution function in the form of the corresponding random variable <code>&lt;:Distributions.ContinuousUnivariateDistribution</code> from <code>Distributions.jl</code>. You may then compute :      - The cdf via <code>Distributions.cdf</code>     - The pdf via <code>Distributions.pdf</code> and the logpdf via <code>Distributions.logpdf</code>     - Samples from the distribution via <code>rand(X,n)</code></p><p>References:      - Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581     - McNeil, Alexander J., and Johanna Nešlehová. &quot;Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions.&quot; (2009): 3059-3097.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/ffb98220fedf75a3b1170b7bcf7916bc8ec07890/src/WilliamsonTransforms.jl#L64-L89">source</a></section></article></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Monday 12 February 2024 22:34">Monday 12 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/91176150bf74b39d9eec4d35eac4e2f022f88fe5/src/WilliamsonTransforms.jl#L3-L47">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="WilliamsonTransforms.𝒲" href="#WilliamsonTransforms.𝒲"><code>WilliamsonTransforms.𝒲</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">𝒲(X,d)(x)</code></pre><p>Computes the Williamson d-transform of the random variable X, taken at point x.</p><p>For a univariate non-negative random variable <span>$X$</span>, with cumulative distribution function <span>$F$</span> and an integer <span>$d\ge 2$</span>, the Williamson-d-transform of <span>$X$</span> is the real function supported on <span>$[0,\infty[$</span> given by:</p><p class="math-container">\[\phi(t) = 𝒲_{d}(X)(t) = \int_{t}^{\infty} \left(1 - \frac{t}{x}\right)^{d-1} dF(x) = \mathbb E\left( (1 - \frac{t}{X})^{d-1}_+\right) \mathbb 1_{t &gt; 0} + \left(1 - F(0)\right)\mathbb 1_{t &lt;0}\]</p><p>This function has several properties:      - We have that <span>$\phi(0) = 1$</span> and <span>$\phi(Inf) = 0$</span>     - <span>$\phi$</span> is <span>$d-2$</span> times derivable, and the signs of its derivatives alternates : <span>$\forall k \in 0,...,d-2, (-1)^k \phi^{(k)} \ge 0$</span>.     - <span>$\phi^{(d-2)}$</span> is convex.</p><p>These properties makes this function what is called an <em>archimedean generator</em>, able to generate <em>archimedean copulas</em> in dimensions up to <span>$d$</span>. </p><p>References: </p><ul><li>Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581</li><li>McNeil, Alexander J., and Johanna Nešlehová. &quot;Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions.&quot; (2009): 3059-3097.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/91176150bf74b39d9eec4d35eac4e2f022f88fe5/src/WilliamsonTransforms.jl#L17-L38">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="WilliamsonTransforms.𝒲₋₁" href="#WilliamsonTransforms.𝒲₋₁"><code>WilliamsonTransforms.𝒲₋₁</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">𝒲₋₁(ϕ,d)</code></pre><p>Computes the inverse Williamson d-transform of the d-monotone archimedean generator ϕ. </p><p>A <span>$d$</span>-monotone archimedean generator is a function <span>$\phi$</span> on <span>$\mathbb R_+$</span> that has these three properties:</p><ul><li><span>$\phi(0) = 1$</span> and <span>$\phi(Inf) = 0$</span></li><li><span>$\phi$</span> is <span>$d-2$</span> times derivable, and the signs of its derivatives alternates : <span>$\forall k \in 0,...,d-2, (-1)^k \phi^{(k)} \ge 0$</span>.</li><li><span>$\phi^{(d-2)}$</span> is convex.</li></ul><p>For such a function <span>$\phi$</span>, the inverse Williamson-d-transform of <span>$\phi$</span> is the cumulative distribution function <span>$F$</span> of a non-negative random variable <span>$X$</span>, defined by : </p><p class="math-container">\[F(x) = 𝒲_{d}^{-1}(\phi)(x) = 1 - \frac{(-x)^{d-1} \phi_+^{(d-1)}(x)}{k!} - \sum_{k=0}^{d-2} \frac{(-x)^k \phi^{(k)}(x)}{k!}\]</p><p>We return this cumulative distribution function in the form of the corresponding random variable <code>&lt;:Distributions.ContinuousUnivariateDistribution</code> from <code>Distributions.jl</code>. You may then compute :      - The cdf via <code>Distributions.cdf</code>     - The pdf via <code>Distributions.pdf</code> and the logpdf via <code>Distributions.logpdf</code>     - Samples from the distribution via <code>rand(X,n)</code></p><p>References:      - Williamson, R. E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207. MR0077581     - McNeil, Alexander J., and Johanna Nešlehová. &quot;Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions.&quot; (2009): 3059-3097.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/lrnv/WilliamsonTransforms.jl/blob/91176150bf74b39d9eec4d35eac4e2f022f88fe5/src/WilliamsonTransforms.jl#L64-L89">source</a></section></article></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Monday 12 February 2024 22:40">Monday 12 February 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>