From 40399cf405c30c7dde49042c581b654d04fa72f3 Mon Sep 17 00:00:00 2001
From: Santymax98 <132070198+Santymax98@users.noreply.github.com>
Date: Sun, 19 Nov 2023 12:16:29 -0300
Subject: [PATCH 1/3] adding calculation of the quantile function

I added the quantile function for the purpose of determining the natural marginals of the Archimedean copulas.
---
 src/WilliamsonTransforms.jl | 8 ++++++++
 1 file changed, 8 insertions(+)

diff --git a/src/WilliamsonTransforms.jl b/src/WilliamsonTransforms.jl
index 9b0c116..8d510e1 100644
--- a/src/WilliamsonTransforms.jl
+++ b/src/WilliamsonTransforms.jl
@@ -121,4 +121,12 @@ function Distributions.rand(rng::Distributions.AbstractRNG, d::𝒲₋₁)
 end
 Base.minimum(::𝒲₋₁) = 0.0
 Base.maximum(::𝒲₋₁) = Inf
+end #This end was here before increasing the following function
+
+function Distributions.quantile(d::𝒲₋₁, p::Real)
+    # Validate that p is in the range [0, 1]
+    @assert 0 <= p <= 1
+
+    # Finding the root of the equation F(x) - p = 0 using the root function
+    return Roots.find_zero(x -> (Distributions.cdf(d, x) - p), (0.0, Inf))
 end

From a296b78ea5351958d890c390ede23487251d624a Mon Sep 17 00:00:00 2001
From: Santymax98 <132070198+Santymax98@users.noreply.github.com>
Date: Sun, 19 Nov 2023 12:29:03 -0300
Subject: [PATCH 2/3] adding test

I added a test with the unknown distribution, in this case the calculated quantile is expected to be equal to the quantile of the Earlang distribution
---
 test/testing_the_paper.jl | 21 ++++++++++++++++++++-
 1 file changed, 20 insertions(+), 1 deletion(-)

diff --git a/test/testing_the_paper.jl b/test/testing_the_paper.jl
index 8957114..5edf592 100644
--- a/test/testing_the_paper.jl
+++ b/test/testing_the_paper.jl
@@ -82,4 +82,23 @@ end
     X = 𝒲₋₁(ϕ,d)
     rand(X,100)
     @test true
-end
\ No newline at end of file
+end
+
+@testitem "Quantile test - IndependantCopula, dimension 10" begin
+    using Distributions
+    for d in 3:20
+        X = Erlang(d)
+        ϕ(x) = exp(-x)
+        Xhat = 𝒲₋₁(ϕ, d)
+
+        # Perform 10 tests of the quantile function
+        for _ in 1:10
+            p = rand()
+            x = quantile(X, p)
+            xhat = quantile(Xhat, p)
+
+            # Verify that the difference between the quantiles is small
+            @test abs(x - xhat) <= sqrt(eps(Float64))
+        end
+    end
+end

From 75356314d12ea791b82cb406dd6c83623ba07abc Mon Sep 17 00:00:00 2001
From: Santymax98 <132070198+Santymax98@users.noreply.github.com>
Date: Sun, 19 Nov 2023 14:05:06 -0300
Subject: [PATCH 3/3] moving function

---
 src/WilliamsonTransforms.jl | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/WilliamsonTransforms.jl b/src/WilliamsonTransforms.jl
index 8d510e1..fcb8976 100644
--- a/src/WilliamsonTransforms.jl
+++ b/src/WilliamsonTransforms.jl
@@ -121,8 +121,6 @@ function Distributions.rand(rng::Distributions.AbstractRNG, d::𝒲₋₁)
 end
 Base.minimum(::𝒲₋₁) = 0.0
 Base.maximum(::𝒲₋₁) = Inf
-end #This end was here before increasing the following function
-
 function Distributions.quantile(d::𝒲₋₁, p::Real)
     # Validate that p is in the range [0, 1]
     @assert 0 <= p <= 1
@@ -130,3 +128,6 @@ function Distributions.quantile(d::𝒲₋₁, p::Real)
     # Finding the root of the equation F(x) - p = 0 using the root function
     return Roots.find_zero(x -> (Distributions.cdf(d, x) - p), (0.0, Inf))
 end
+end
+
+