Normal Copula |
Let for , where represents the uniform distribution on the interval. Let be the correlation matrix with parameters satisfying 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 -variate standard normal distribution with mean vector and covariance matrix . That is, the distribution is .
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 -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 is a lower triangular matrix with positive elements on the diagonal. If , then .
To fit a normal copula is to estimate the covariance matrix from an input sample data set. Given a random sample where , the log-likelihood function is
Here is the joint density of the multivariate normal with mean zero and variance , and is the univariate density of the standard normal distribution. Note that the second term is not related to the parameters and, therefore, can be ignored during the optimization. The restriction that is a correlation matrix is very inconvenient, and it is common practice to circumvent this problem by first assuming that has the covariance form. Therefore, can be estimated by
where
This estimate is consistent with the form of a covariance matrix but not necessarily with the form of a correlation matrix. The approximation to the original MLE problem can be obtained using the normalizing operator defined as follows: