# The COPULA Procedure

### Canonical Maximum Likelihood Estimation (CMLE)

In the canonical maximum likelihood estimation (CMLE) method, it is assumed that the sample data , have been transformed into uniform variates , . One commonly used transformation is the nonparametric estimation of the CDF of the marginal distributions, which is closely related to empirical CDF,

where

The transformed data are used as if they had uniform marginal distributions; hence, they are called pseudo-samples. The function is different from the standard empirical CDF in the scalar , which is to ensure that the transformed data cannot be on the boundary of the unit interval . It is clear that

where is the rank among in increasing order.

Let be the density function of a copula , and let be the parameter vector to be estimated. The parameter is estimated by maximum likelihood: