diff --git a/src/ArchimedeanCopula.jl b/src/ArchimedeanCopula.jl
index 33e141a5..dfe6a886 100644
--- a/src/ArchimedeanCopula.jl
+++ b/src/ArchimedeanCopula.jl
@@ -88,8 +88,10 @@ end
 ϕ⁻¹(C::ArchimedeanCopula{d},x) where d = Roots.find_zero(t -> ϕ(C,t) - x, (0.0, Inf))
 
 williamson_dist(C::ArchimedeanCopula{d}) where d = WilliamsonTransforms.𝒲₋₁(t -> ϕ(C,t),d)
-τ(C::ArchimedeanCopula)  = 4*Distributions.expectation(r -> ϕ(C,r), williamson_dist(C)) - 1
-
+function τ(C::ArchimedeanCopula)  
+    @show C
+    return 4*Distributions.expectation(r -> ϕ(C,r), williamson_dist(C)) - 1
+end
 
 function _archi_rand!(rng,C::ArchimedeanCopula{d},R,x) where d
     # x is assumed to already be random exponentials produced by Random.randexp
diff --git a/src/ArchimedeanCopulas/AMHCopula.jl b/src/ArchimedeanCopulas/AMHCopula.jl
index 0aabc3ac..c00d881f 100644
--- a/src/ArchimedeanCopulas/AMHCopula.jl
+++ b/src/ArchimedeanCopulas/AMHCopula.jl
@@ -32,16 +32,25 @@ end
 ϕ(  C::AMHCopula,t) = (1-C.θ)/(exp(t)-C.θ)
 ϕ⁻¹(  C::AMHCopula,t) = log(C.θ + (1-C.θ)/t)
 
-τ(C::AMHCopula) = 1 - 2(C.θ+(1-C.θ)^2*log(1-C.θ))/(3C.θ^2) # no closed form inverse...
+τ(C::AMHCopula) = _amh_tau_f(C.θ) # no closed form inverse...
+
+_amh_tau_f(θ) = θ == 1 ? 1/3 : 1 - 2(θ+(1-θ)^2*log1p(-θ))/(3θ^2)
+
+# if θ = -1, we obtain (5 -8*log(2))/3 
+
 function τ⁻¹(::Type{AMHCopula},τ)
     if τ == zero(τ)
         return τ
     end
     if τ > 1/3
         @warn "AMHCopula cannot handle kendall tau's greater than 1/3. We capped it to 1/3."
-        return 1
+        return one(τ)
+    end
+    if τ < (5 - 8*log(2))/3
+        @warn "AMHCopula cannot handle kendall tau's smaller than (5- 8ln(2))/3 (approx -0.1817). We capped it to this value."
+        return -one(τ)
     end
-    return Roots.find_zero(θ -> 1 - 2(θ+(1-θ)^2*log(1-θ))/(3θ^2) - τ, (-1.0, 1.0))
+    return Roots.find_zero(θ -> _amh_tau_f(θ) - τ, (-one(τ), one(τ)))
 end
 williamson_dist(C::AMHCopula{d,T}) where {d,T} = C.θ >= 0 ? WilliamsonFromFrailty(1 + Distributions.Geometric(1-C.θ),d) : WilliamsonTransforms.𝒲₋₁(t -> ϕ(C,t),d)
 
diff --git a/src/ArchimedeanCopulas/FrankCopula.jl b/src/ArchimedeanCopulas/FrankCopula.jl
index 01354f73..6a015961 100644
--- a/src/ArchimedeanCopulas/FrankCopula.jl
+++ b/src/ArchimedeanCopulas/FrankCopula.jl
@@ -58,6 +58,9 @@ function τ⁻¹(::Type{FrankCopula},τ)
     if τ == zero(τ)
         return τ
     end
+    if abs(τ==1)
+        return Inf * τ
+    end
     x₀ = (1-τ)/4
     return Roots.fzero(x -> (1-D₁(x))/x - x₀, 1e-4, Inf)
 end
diff --git a/src/ArchimedeanCopulas/GumbelBarnettCopula.jl b/src/ArchimedeanCopulas/GumbelBarnettCopula.jl
index 92d04473..aec315a8 100644
--- a/src/ArchimedeanCopulas/GumbelBarnettCopula.jl
+++ b/src/ArchimedeanCopulas/GumbelBarnettCopula.jl
@@ -39,22 +39,23 @@ end
 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, _ = gsl_integration_qags(f, 0.0, 1.0, [C.θ], 1e-7, 1000)
+    # result = 
+    # result, _ = GSL.integration_qags(f, 0.0, 1.0, [C.θ], 1e-7, 1000)
     
     return 1+4*result
 end
-function τ⁻¹(::Type{GumbelBarnettCopula}, τ)
-    if τ == zero(τ)
-        return τ
+function τ⁻¹(::Type{GumbelBarnettCopula}, tau)
+    if tau == zero(tau)
+        return tau
     end
     
     # Define an anonymous function that takes a value x and computes τ 
     #for a GumbelBarnettCopula with θ = x
-    τ_func(x) = τ(GumbelBarnettCopula{d, Float64}(x))
+    τ_func(x) = τ(GumbelBarnettCopula(2,x))
     
     # Use the bisection method to find the root
-    x = Roots.find_zero(x -> τ_func(x) - τ, (0.0, 1.0))    
+    x = Roots.find_zero(x -> τ_func(x) - tau, (0.0, 1.0))    
     return x
 end
\ No newline at end of file
diff --git a/src/ArchimedeanCopulas/GumbelCopula.jl b/src/ArchimedeanCopulas/GumbelCopula.jl
index 73c775b9..43e2cbb4 100644
--- a/src/ArchimedeanCopulas/GumbelCopula.jl
+++ b/src/ArchimedeanCopulas/GumbelCopula.jl
@@ -22,7 +22,8 @@ struct GumbelCopula{d,T} <: ArchimedeanCopula{d}
     θ::T
     function GumbelCopula(d,θ)
         if θ < 1
-            throw(ArgumentError("Theta must be greater than 1"))
+            @show θ
+            throw(ArgumentError("Theta must be greater than or equal to 1"))
         elseif θ == 1
             return IndependentCopula(d)
         elseif θ == Inf
@@ -35,7 +36,18 @@ end
 ϕ(  C::GumbelCopula,       t) = exp(-t^(1/C.θ))
 ϕ⁻¹(C::GumbelCopula,       t) = (-log(t))^C.θ
 τ(C::GumbelCopula) = ifelse(isfinite(C.θ), (C.θ-1)/C.θ, 1)
-τ⁻¹(::Type{GumbelCopula},τ) =ifelse(τ == 1, Inf, 1/(1-τ))
+function τ⁻¹(::Type{GumbelCopula},τ) 
+    if τ == 1
+        return Inf
+    else
+        θ = 1/(1-τ)
+        if θ < 1
+            @warn "GumbelCopula cannot handle negative kendall tau's, returning independence.."
+            return 1
+        end
+        return θ
+    end
+end
 williamson_dist(C::GumbelCopula{d,T}) where {d,T} = WilliamsonFromFrailty(AlphaStable(α = 1/C.θ, β = 1,scale = cos(π/(2C.θ))^C.θ, location = (C.θ == 1 ? 1 : 0)), d)
 
 
diff --git a/src/ArchimedeanCopulas/InvGaussianCopula.jl b/src/ArchimedeanCopulas/InvGaussianCopula.jl
index 69a2513c..4da22a7e 100644
--- a/src/ArchimedeanCopulas/InvGaussianCopula.jl
+++ b/src/ArchimedeanCopulas/InvGaussianCopula.jl
@@ -28,6 +28,8 @@ struct InvGaussianCopula{d,T} <: ArchimedeanCopula{d}
             throw(ArgumentError("Theta must be non-negative."))
         elseif θ == 0
             return IndependentCopula(d)
+        elseif isinf(θ)
+            throw(ArgumentError("Theta cannot be infinite"))
         else
             return new{d,typeof(θ)}(θ)
         end
@@ -51,7 +53,7 @@ function τ⁻¹(::Type{InvGaussianCopula}, τ)
     end
 
     # Define an anonymous function that takes a value x and computes τ for an InvGaussianCopula copula with θ = x
-    τ_func(x) = τ(InvGaussianCopula{d, Float64}(x))
+    τ_func(x) = τ(InvGaussianCopula{2, Float64}(x))
 
     # Set an initial value for x₀ (adjustable)
     x₀ = (1-τ)/4
diff --git a/src/ArchimedeanCopulas/JoeCopula.jl b/src/ArchimedeanCopulas/JoeCopula.jl
index c8e08ec1..bffa7f70 100644
--- a/src/ArchimedeanCopulas/JoeCopula.jl
+++ b/src/ArchimedeanCopulas/JoeCopula.jl
@@ -34,7 +34,20 @@ struct JoeCopula{d,T} <: ArchimedeanCopula{d}
 end
 ϕ(  C::JoeCopula,          t) = 1-(-expm1(-t))^(1/C.θ)
 ϕ⁻¹(C::JoeCopula,          t) = -log1p(-(1-t)^C.θ)
-τ(C::JoeCopula) = 1 - 4sum(1/(k*(2+k*C.θ)*(C.θ*(k-1)+2)) for k in 1:1000) # 446 in R copula. 
+_joe_tau_f(θ) = 1 - 4sum(1/(k*(2+k*θ)*(θ*(k-1)+2)) for k in 1:1000) # 446 in R copula. 
+τ(C::JoeCopula) = _joe_tau_f(C.θ)
+function τ⁻¹(::Type{JoeCopula},τ) 
+    if τ == 1
+        return Inf
+    elseif τ == 0
+        return 1
+    elseif τ < 0
+        @warn "JoeCoula cannot handle negative kendall taus, we return the independence..."
+        return one(τ)
+    else
+        return Roots.find_zero(θ -> _joe_tau_f(θ) - τ, (one(τ),τ*Inf))
+    end
+end
 williamson_dist(C::JoeCopula{d,T}) where {d,T} = WilliamsonFromFrailty(Sibuya(1/C.θ), d)
 
 
diff --git a/src/Copula.jl b/src/Copula.jl
index b54b6974..e4edc54c 100644
--- a/src/Copula.jl
+++ b/src/Copula.jl
@@ -9,10 +9,13 @@ Base.length(::Copula{d}) where d = d
 # Base.eltype
 # τ, τ⁻¹
 # Base.eltype 
-function Distributions.cdf(C::Copula{d},u) where d
+function Distributions.cdf(C::Copula{d},u::AbstractVector) where d
     length(u) != length(C) && throw(ArgumentError("Dimension mismatch between copula and input vector"))
     return _cdf(C,u)
 end
+function Distributions.cdf(C::Copula{d},A::AbstractMatrix) where d
+    return [Distributions.cdf(C,u) for u in eachcol(A)]
+end
 function _cdf(C::CT,u) where {CT<:Copula}
     f(x) = Distributions.pdf(C,x)
     z = zeros(eltype(u),length(C))
diff --git a/src/MiscellaneousCopulas/MCopula.jl b/src/MiscellaneousCopulas/MCopula.jl
index 0f53a9c5..463fb3ce 100644
--- a/src/MiscellaneousCopulas/MCopula.jl
+++ b/src/MiscellaneousCopulas/MCopula.jl
@@ -18,6 +18,9 @@ The two Frechet-Hoeffding bounds are also Archimedean copulas, although this lin
 struct MCopula{d} <: Copula{d} end
 MCopula(d) = MCopula{d}()
 _cdf(::MCopula{d},u) where {d} = minimum(u)
+function Distributions._logpdf(C::MCopula, u)
+    return all(u == u[1]) ? zero(eltype(u)) : eltype(u)(-Inf)
+end
 function Distributions._rand!(rng::Distributions.AbstractRNG, ::MCopula{d}, x::AbstractVector{T}) where {d,T<:Real}
     x .= rand(rng)
 end
diff --git a/src/MiscellaneousCopulas/WCopula.jl b/src/MiscellaneousCopulas/WCopula.jl
index 2e0f2e62..a529de83 100644
--- a/src/MiscellaneousCopulas/WCopula.jl
+++ b/src/MiscellaneousCopulas/WCopula.jl
@@ -16,6 +16,10 @@ The two Frechet-Hoeffding bounds are also Archimedean copulas, although this lin
 struct WCopula{d} <: Copula{d} end
 WCopula(d) = WCopula{d}()
 _cdf(::WCopula{d},u) where {d} = max(1 + sum(u)-d,0)
+function Distributions._logpdf(C::WCopula{d}, u) where {d}
+    @assert d == 2
+    return u[1] + u[2] == 1 ? zero(eltype(u)) : eltype(u)(-Inf)
+end
 function Distributions._rand!(rng::Distributions.AbstractRNG, ::WCopula{d}, x::AbstractVector{T}) where {d,T<:Real}
     @assert d==2
     x[1] = rand(rng)
diff --git a/test/archimedean_tests.jl b/test/archimedean_tests.jl
index f142d7f6..3f51f2d7 100644
--- a/test/archimedean_tests.jl
+++ b/test/archimedean_tests.jl
@@ -1,8 +1,6 @@
 
 @testitem "Test of τ ∘ τ_inv bijection" begin
     using Random
-    using StableRNGs
-    rng = StableRNG(123)
     taus = [0.0, 0.1, 0.5, 0.9, 1.0]
 
     for T in (
@@ -21,79 +19,138 @@
     end
 end
 
+# For each archimedean, we test: 
+# - The sampling of a dataset
+# - evaluation of pdf and cdf
+# - fitting on the dataset. 
 
-@testitem "AMHCopula - Test sampling all cases" begin
+@testitem "AMHCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     for d in 2:10
         for θ ∈ [-1.0,-rand(rng),0.0,rand(rng)]
-            rand(rng,AMHCopula(d,θ),10)
+            C = AMHCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            fit(AMHCopula,data)
         end
     end
     @test true
 end
-@testitem "ClaytonCopula - Test sampling all cases" begin
+@testitem "ClaytonCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
-    rand(rng,ClaytonCopula(2,-1),10)
+    C = ClaytonCopula(2,-1)
+    data = rand(rng,C,100)
+    @test all(pdf(C,data) .>= 0)
+    @test all(0 .<= cdf(C,data) .<= 1)
+    fit(ClaytonCopula,data)
     for d in 2:10
         for θ ∈ [-1/(d-1) * rand(rng),0.0,-log(rand(rng)), Inf]
-            rand(rng,ClaytonCopula(d,θ),10)
+            C = ClaytonCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            fit(ClaytonCopula,data)
         end
     end
     @test true
 end
-@testitem "FrankCopula - Test sampling all cases" begin
+@testitem "FrankCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     rand(rng,FrankCopula(2,-Inf),10)
     rand(rng,FrankCopula(2,log(rand(rng))),10)
+
+    C = FrankCopula(2,-Inf)
+    data = rand(rng,C,100)
+    @test all(pdf(C,data) .>= 0)
+    @test all(0 .<= cdf(C,data) .<= 1)
+    @test_broken fit(FrankCopula,data)
+
+    C = FrankCopula(2,log(rand(rng)))
+    data = rand(rng,C,100)
+    @test all(pdf(C,data) .>= 0)
+    @test all(0 .<= cdf(C,data) .<= 1)
+    @test_broken fit(FrankCopula,data)
+
+
     for d in 2:10
         for θ ∈ [0.0,rand(rng),1.0,-log(rand(rng)), Inf]
-            rand(rng,FrankCopula(d,θ),10)
+            C = FrankCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            @test_broken fit(FrankCopula,data)
         end
     end
     @test true
 end
-@testitem "GumbelCopula - Test sampling all cases" begin
+@testitem "GumbelCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     for d in 2:10
         for θ ∈ [1.0,1-log(rand(rng)), Inf]
-            rand(rng,GumbelCopula(d,θ),10)
+            @show d,θ
+            C = GumbelCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            fit(GumbelCopula,data)
         end
     end
     @test true
 end
-@testitem "JoeCopula - Test sampling all cases" begin
+@testitem "JoeCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     for d in 2:10
         for θ ∈ [1.0,1-log(rand(rng)), Inf]
-            rand(rng,JoeCopula(d,θ),10)
+            C = JoeCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            fit(JoeCopula,data)
         end
     end
     @test true
 end
 
-@testitem "GumbelBarnettCopula - Test sampling all cases" begin
+@testitem "GumbelBarnettCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     for d in 2:10
         for θ ∈ [0.0,rand(rng),1.0]
-            rand(rng,GumbelBarnettCopula(d,θ),10)
+            C = GumbelBarnettCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            @test_broken fit(GumbelBarnettCopula,data)
         end
     end
     @test true
 end
 
-@testitem "InvGaussianCopula - Test sampling all cases" begin
+@testitem "InvGaussianCopula - sampling,pdf,cdf,fit" begin
     using StableRNGs
+    using Distributions
     rng = StableRNG(123)
     for d in 2:10
-        for θ ∈ [rand(rng),1.0, Inf]
-            rand(rng,InvGaussianCopula(d,θ),10)
+        for θ ∈ [rand(rng),1.0, -log(rand(rng))]
+            @show d,θ
+            C = InvGaussianCopula(d,θ)
+            data = rand(rng,C,100)
+            @test all(pdf(C,data) .>= 0)
+            @test all(0 .<= cdf(C,data) .<= 1)
+            @test_broken fit(InvGaussianCopula,data)
         end
     end
     @test true
-end
\ No newline at end of file
+end