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| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Liouville Copulas · Copulas.jl</title><meta name="title" content="Liouville Copulas · Copulas.jl"/><meta property="og:title" content="Liouville Copulas · Copulas.jl"/><meta property="twitter:title" content="Liouville Copulas · Copulas.jl"/><meta name="description" content="Documentation for Copulas.jl."/><meta property="og:description" content="Documentation for Copulas.jl."/><meta property="twitter:description" content="Documentation for Copulas.jl."/><meta property="og:url" content="https://lrnv.github.io/Copulas.jl/Liouville/"/><meta property="twitter:url" content="https://lrnv.github.io/Copulas.jl/Liouville/"/><link rel="canonical" href="https://lrnv.github.io/Copulas.jl/Liouville/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" 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id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../"><img class="docs-light-only" src="../assets/logo.svg" alt="Copulas.jl logo"/><img class="docs-dark-only" src="../assets/logo-dark.svg" alt="Copulas.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../">Copulas.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Manual</span><ul><li><a class="tocitem" href="../getting_started/">Getting Started</a></li><li><a class="tocitem" href="../sklar/">Sklar's Distributions</a></li><li><a class="tocitem" href="../elliptical/generalities/">Elliptical Copulas</a></li><li><a class="tocitem" href="../archimedean/generalities/">Archimedean Copulas</a></li><li class="is-active"><a class="tocitem" href>Liouville Copulas</a></li><li><a class="tocitem" href="../extremevalue/generalities/">Extreme Value Copulas</a></li><li><a class="tocitem" href="../empirical/generalities/">Empirical Copulas</a></li><li><a class="tocitem" href="../dependence_measures/">Dependence measures</a></li></ul></li><li><span class="tocitem">Bestiary</span><ul><li><a class="tocitem" href="../elliptical/available_models/">Elliptical Copulas</a></li><li><a class="tocitem" href="../archimedean/available_models/">Archimedean Generators</a></li><li><a class="tocitem" href="../extremevalue/available_models/">Extreme Values Copulas</a></li><li><a class="tocitem" href="../empirical/available_models/">Empirical Copulas</a></li><li><a class="tocitem" href="../miscellaneous/">Other Copulas</a></li><li><a class="tocitem" href="../transformations/">Transformed Copulas</a></li></ul></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/lambda_viz/">Empirical Kendall function and Archimedean's λ function.</a></li><li><a class="tocitem" href="../examples/lossalae/">Loss-Alae fitting example.</a></li><li><a class="tocitem" href="../examples/fitting_sklar/">Fitting compound distributions</a></li><li><a class="tocitem" href="../examples/ifm1/">Influence of the method of estimation</a></li><li><a class="tocitem" href="../examples/turing/">Bayesian inference with <code>Turing.jl</code></a></li><li><a class="tocitem" href="../examples/other_usecases/">Other known use cases.</a></li></ul></li><li><a class="tocitem" href="../dev_roadmap/">Dev Roadmap</a></li><li><a class="tocitem" href="../idx/">Package Index</a></li><li><a class="tocitem" href="../references/">References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Liouville Copulas</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Liouville Copulas</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/lrnv/Copulas.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/lrnv/Copulas.jl/blob/main/docs/src/Liouville.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Liouville-Copulas"><a class="docs-heading-anchor" href="#Liouville-Copulas">Liouville Copulas</a><a id="Liouville-Copulas-1"></a><a class="docs-heading-anchor-permalink" href="#Liouville-Copulas" title="Permalink"></a></h1><div class="admonition is-info"><header class="admonition-header">Not merged yet !</header><div class="admonition-body"><p>Liouville copulas are coming in this PR : https://github.com/lrnv/Copulas.jl/pull/83, but this is not merged yet. </p></div></div><p>Archimedean copulas have been widely used in the literature due to their nice decomposition properties and easy parametrization. The interested reader can refer to the extensive literature [<a href="../references/#hofert2010">24</a>–<a href="../references/#dibernardino2013a">30</a>, <a href="../references/#dibernardino2013a">30</a>–<a href="../references/#spreeuw2014">33</a>] on Archimedean copulas, their nesting extensions and most importantly their estimation. </p><p>One major drawback of the Archimedean family is that these copulas have exchangeable marginals (i.e., <span>$C(\bm u) = C(\mathrm{p}(\bm u))$</span> for any permutation <span>$p(\bm u)$</span> of <span>$u_1,...,u_d$</span>): the dependence structure is symmetric, which might not be a wanted property. However, from the Radial-simplex expression, we can easily extrapolate a little and take for <span>$\bm S$</span> a non-uniform distribution on the simplex. </p><p>Liouville's copulas share many properties with Archimedean copulas, but are not exchangeable anymore. This is an easy way to produce non-exchangeable dependence structures. See [<a href="../references/#cote2019">15</a>] for a practical use of this property.</p><p>Note that Dirichlet distributions are constructed as <span>$\bm S = \frac{\bm G}{\langle \bm 1, \bm G\rangle}$</span>, where <span>$\bm G$</span> is a vector of independent Gamma distributions with unit scale (and potentially different shapes: taking all shapes equal yields the Archimedean case). </p><div class="citation noncanonical"><dl><dt>[15]</dt><dd><div>M.-P. Côté and C. Genest. <em>Dependence in a Background Risk Model</em>. Journal of Multivariate Analysis <strong>172</strong>, 28–46 (2019).</div></dd><dt>[24]</dt><dd><div>M. Hofert. <em>Sampling Nested Archimedean Copulas with Applications to CDO Pricing</em>. Ph.D. Thesis, Universität Ulm (2010).</div></dd><dt>[25]</dt><dd><div>M. Hofert and D. Pham. <em>Densities of Nested Archimedean Copulas</em>. Journal of Multivariate Analysis <strong>118</strong>, 37–52 (2013).</div></dd><dt>[26]</dt><dd><div>A. J. McNeil and J. Nešlehová. <em>From Archimedean to Liouville Copulas</em>. Journal of Multivariate Analysis <strong>101</strong>, 1772–1790 (2010).</div></dd><dt>[27]</dt><dd><div>H. Cossette, S.-P. Gadoury, E. Marceau and I. Mtalai. <em>Hierarchical Archimedean Copulas through Multivariate Compound Distributions</em>. Insurance: Mathematics and Economics <strong>76</strong>, 1–13 (2017).</div></dd><dt>[28]</dt><dd><div>H. Cossette, E. Marceau, I. Mtalai and D. Veilleux. <em>Dependent Risk Models with Archimedean Copulas: A Computational Strategy Based on Common Mixtures and Applications</em>. Insurance: Mathematics and Economics <strong>78</strong>, 53–71 (2018).</div></dd><dt>[29]</dt><dd><div>C. Genest, J. Nešlehová and J. Ziegel. <em>Inference in Multivariate Archimedean Copula Models</em>. TEST <strong>20</strong>, 223–256 (2011).</div></dd><dt>[30]</dt><dd><div>E. Di Bernardino and D. Rulliere. <em>On Certain Transformations of Archimedean Copulas: Application to the Non-Parametric Estimation of Their Generators</em>. Dependence Modeling <strong>1</strong>, 1–36 (2013).</div></dd><dt>[31]</dt><dd><div>E. Di Bernardino and D. Rullière. <em>On an Asymmetric Extension of Multivariate Archimedean Copulas Based on Quadratic Form</em>. Dependence Modeling <strong>4</strong> (2016).</div></dd><dt>[32]</dt><dd><div>K. Cooray. <em>Strictly Archimedean Copulas with Complete Association for Multivariate Dependence Based on the Clayton Family</em>. Dependence Modeling <strong>6</strong>, 1–18 (2018).</div></dd><dt>[33]</dt><dd><div>J. Spreeuw. <em>Archimedean Copulas Derived from Utility Functions</em>. Insurance: Mathematics and Economics <strong>59</strong>, 235–242 (2014).</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../archimedean/generalities/">« Archimedean Copulas</a><a class="docs-footer-nextpage" href="../extremevalue/generalities/">Extreme Value Copulas »</a><div class="flexbox-break"></div><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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Friday 20 September 2024 09:02">Friday 20 September 2024</span>. 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