Elliptical Copulas

GaussianCopula{d,MT}

Fields:

• Σ::MT - covariance matrix

Constructor

GaussianCopula(Σ)

The Gaussian Copula is the copula of a Multivariate normal distribution. It is constructed as:

\[C(\mathbf{x}; \boldsymbol{\Sigma}) = F_{\Sigma}(F_{\Sigma,i}^{-1}(x_i),i\in 1,...d)\]

where $F_{\Sigma}$ is a cdf of a gaussian random vector and $F_{\Sigma,i}$ is the ith marginal cdf, while $\Sigma$ is the covariance matrix.

It can be constructed in Julia via:

C = GaussianCopula(Σ)

You can sample it, compute pdf and cdf, or even fit the distribution via:

u = rand(C,1000)
Random.rand!(C,u) # other calling syntax for rng.
pdf(C,u) # to get the density
cdf(C,u) # to get the distribution function 
Ĉ = fit(GaussianCopula,u) # to fit on the sampled data. 

GaussianCopulas have a special case:

• When isdiag(Σ), the constructor returns an IndependentCopula(d)

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

TCopula{d,MT}

Fields:

• df::Int - number of degree of freedom
• Σ::MT - covariance matrix

Constructor

TCopula(df,Σ)

The Student's T Copula is the copula of a Multivariate Student distribution. It is constructed as :

\[C(\mathbf{x}; \boldsymbol{n,\Sigma}) = F_{n,\Sigma}(F_{n,\Sigma,i}^{-1}(x_i),i\in 1,...d)\]

where $F_{n,\Sigma}$ is a cdf of a multivariate student random vector with covariance matrix $\Sigma$ and $n$ degrees of freedom. and F_{n,\Sigma,i} is the ith marignal cdf.

It can be constructed in Julia via:

C = TCopula(2,Σ)

You can sample it, compute pdf and cdf, or even fit the distribution via:

u = rand(C,1000)
Random.rand!(C,u) # other calling syntax for rng.
pdf(C,u) # to get the density
cdf(C,u) # to get the distribution function 
Ĉ = fit(TCopula,u) # to fit on the sampled data. 

Except that currently it does not work since fit(Distributions.MvTDist,data) does not dispatch.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.