Gumbel copula probability density incorrect for theta = 1

[LiZhongyangLi]

Environment Details

• Copulas version: 0.5.1
• Python version: 3.6.3
• Operating System: CentOS Linux release 7.9.2009

Error Description

When theta is 1, Gumbel copula should be reduced to the independence copula with a culumative distribution equal to uv and a probability density equal constant 1. In Copulas/copulas/bivariate/gumbel.py line 46 this is implemented as the product of u and v:

  if self.theta == 1:
            return np.multiply(U, V)

Also in the comment the class is mistakenly described as the Clayton copula:

"""Class for clayton copula model."""

Steps to reproduce

from copulas.bivariate import gumbel
g = gumbel.Gumbel()
g.theta = 1
g.probability_density(np.array([0.3,0.2]).reshape(1,2))

array([0.06])