Canonical Maximum Likelihood Estimation (CMLE) :: SAS/ETS(R) 12.3 User's Guide

Canonical Maximum Likelihood Estimation (CMLE)

In the CMLE estimation method, it is assumed that the sample data $\pmb x_ i = ( x_{i1}, x_{i2}, \ldots , x_{im} )^\top$ , $i=1,\ldots ,n$ have been transformed into uniform variates $\boldsymbol {\hat{u}}_ i = (\hat{u}_{i1}, \ldots , \hat{u}_{im})$ , $i=1,\ldots ,n$ . One commonly used transformation is the nonparametric estimation of the CDF of the marginal distributions, which is closely related to empirical CDF,

$\hat{u}_{i,j}= \hat{F}_{j,n}(x_{i,j})$

where

$\hat{F}_{j,n}(x) = \frac{1}{n+1}\sum _{i=1}^ n I_{[x_{i,j}\le x]}$

The transformed data $\hat{u}_{i,j}$ are used as if they had uniform marginal distributions; hence, they are called pseudo-samples. The function $\hat{F}_{j,n}$ 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

$\hat{u}_{i,j} = \frac{1}{n+1} \textrm{rank}(x_{i,j})$

where $\textrm{rank}(x_{i,j})$ is the rank among $i=1,\ldots ,n$ in increasing order.

Let $c(u_1, u_2,{\ldots } u_ m; {\theta })$ be the density function of a copula $C(u_1, u_2,{\ldots } u_ m; {\theta })$ , and let $\theta$ be the parameter vector to be estimated. The parameter $\theta$ is estimated by maximum likelihood:

$\hat{\theta } = \arg \max _{{\theta }\in {\Theta }} \sum _{i=1}^{n} \log c(\hat{u}_{i1}, {\ldots }, \hat{u}_{im}; {\theta })$

The COPULA Procedure (Experimental)

Canonical Maximum Likelihood Estimation (CMLE)