Archimedeans

src/ArchimedeanCopulas/GumbelBarnettCopula.jl

@@ -37,12 +37,8 @@ end
 ϕ(  C::GumbelBarnettCopula,       t) = exp((1-exp(t))/C.θ)
 ϕ⁻¹(C::GumbelBarnettCopula,       t) = log(1-C.θ*log(t))
 function τ(C::GumbelBarnettCopula)
-    # Define the function to integrate
-    f(x) = -x * (1 - C.θ * log(x)) * log(1 - C.θ * log(x)) / C.θ
-    result = Distributions.expectation(f,Distributions.Uniform(0,1))
-    # Calculate the integral using GSL
-    # result = 
-    # result, _ = GSL.integration_qags(f, 0.0, 1.0, [C.θ], 1e-7, 1000)
+    # Use a numerical integration method to obtain tau
+    result, _ = quadgk(x -> -((x-C.θ*x*log(x))*log(1-C.θ*ln(x))/C.θ), 0, 1)
 
     return 1+4*result
 end
@@ -58,4 +54,4 @@ function τ⁻¹(::Type{GumbelBarnettCopula}, tau)
     # Use the bisection method to find the root
     x = Roots.find_zero(x -> τ_func(x) - tau, (0.0, 1.0))    
     return x
-end
+end

src/ArchimedeanCopulas/InvGaussianCopula.jl

@@ -38,12 +38,11 @@ end
 
 ϕ(  C::InvGaussianCopula,       t) = exp((1-sqrt(1+2*((C.θ)^(2))*t))/C.θ)
 ϕ⁻¹(C::InvGaussianCopula,       t) = ((1-C.θ*log(t))^(2)-1)/(2*(C.θ)^(2))
+
 function τ(C::InvGaussianCopula)
-    # Define the function to integrate
-    f(x) = -x * (((1 - C.θ * log(x))^2 - 1) / (2 * C.θ * (1 - C.θ * log(x))))
 
     # Calculate the integral using an appropriate numerical integration method
-    result, _ = gsl_integration_qags(f, 0.0, 1.0, [C.θ], 1e-7, 1000)
+    result, _ = quadgk( x -> (x*((1-C.θ*log(x))^2-1))/(-2*C.θ*(1-C.θ*log(x))),0,1)
 
     return 1+4*result
 end

test/archimedean_tests.jl

@@ -1,17 +1,16 @@
-
 @testitem "Test of τ ∘ τ_inv bijection" begin
     using Random
     taus = [0.0, 0.1, 0.5, 0.9, 1.0]
 
     for T in (
         # AMHCopula,
-        ClaytonCopula,
+        #ClaytonCopula,
         # FrankCopula,
-        GumbelCopula,
+        #GumbelCopula,
         # IndependentCopula,
         # JoeCopula,
-        # GumbelBarnettCopula,
-        # InvGaussianCopula
+         GumbelBarnettCopula,
+         InvGaussianCopula
     )
         for τ in taus
             @test Copulas.τ(T(2,Copulas.τ⁻¹(T,τ))) ≈ τ