HPCOPULA Procedure
Normal Copula
Let 
for 
, where 
represents the uniform distribution on the 
interval. Let 
be the correlation matrix, where 
parameters satisfy the positive semidefiniteness constraint. The normal copula can be written as
where 
is the distribution function of a standard normal random variable and 
is the m-variate standard normal distribution with mean vector 0 and covariance matrix . That is, the distribution is 
.
Simulation
For the normal copula, the input of the simulation is the correlation matrix . The normal copula can be simulated by the following steps, in which 
denotes one random draw from the copula:
Generate a multivariate normal vector
, where is an m-dimensional correlation matrix. Transform the vector
into 
, where is the distribution function of univariate standard normal.
The first step can be achieved by Cholesky decomposition of the correlation matrix 
, where L is a lower triangular matrix with positive elements on the diagonal. If 
, then 
.