move liouville

lrnv · Sep 22, 2024 · 748fe61 · 748fe61
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diff --git a/docs/make.jl b/docs/make.jl
@@ -26,8 +26,7 @@ makedocs(;
             "Getting Started" => "getting_started.md",
             "Sklar's Distributions" => "sklar.md",
             "Elliptical Copulas" => "elliptical/generalities.md",
-            "Archimedean Copulas" => "archimedean/generalities.md",
-            "Liouville Copulas" => "Liouville.md",
+            "Archimedean Generators" => "archimedean/generalities.md",
             "Extreme Value Copulas" => "extremevalue/generalities.md",
             "Empirical Copulas" => "empirical/generalities.md",
             "Vines Copulas" => "Vines.md",

diff --git a/docs/src/Liouville.md b/docs/src/Liouville.md
diff --git a/docs/src/archimedean/generalities.md b/docs/src/archimedean/generalities.md
@@ -211,12 +211,27 @@ for which the corresponding distribution is known but has no particular name, th
 ArchimedeanCopula
 ```
 
-
 <!-- 
 TODO: Make a few graphs of bivariate archimedeans pdfs and cdfs. And provide a few more standard tools for these copulas ? 
 -->
 
+# Liouville Copulas
+
+!!! todo "Not merged yet !"
+    Liouville copulas are coming in this PR : https://github.com/lrnv/Copulas.jl/pull/83, but the work is not finished. 
+
+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 [hofert2010,hofert2013a,mcneil2010,cossette2017,cossette2018,genest2011a,dibernardino2013a,dibernardino2013a,dibernardino2016,cooray2018,spreeuw2014](@cite) on Archimedean copulas, their nesting extensions and most importantly their estimation. 
+
+One major drawback of the Archimedean family is that these copulas have exchangeable marginals (i.e., $C(\bm u) = C(\mathrm{p}(\bm u))$ for any permutation $p(\bm u)$ of $u_1,...,u_d$): 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 $\bm S$ a non-uniform distribution on the simplex. 
+
+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 [cote2019](@cite) for a practical use of this property.
+
+Note that Dirichlet distributions are constructed as $\bm S = \frac{\bm G}{\langle \bm 1, \bm G\rangle}$, where $\bm G$ is a vector of independent Gamma distributions with unit scale (and potentially different shapes: taking all shapes equal yields the Archimedean case). 
+
+
+
+
 ```@bibliography
-Pages = ["generalities.md"]
+Pages = [@__FILE__]
 Canonical = false
 ```