COPULA Procedure

Canonical Maximum Likelihood Estimation (CMLE)

In the canonical maximum likelihood estimation (CMLE) method, it is assumed that the sample data x x Subscript i Baseline equals left-parenthesis x Subscript i Baseline 1 Baseline comma x Subscript i Baseline 2 Baseline comma ellipsis comma x Subscript i m Baseline right-parenthesis Superscript down-tack, i equals 1 comma ellipsis comma n, have been transformed into uniform variates ModifyingAbove bold-italic u With bold caret Subscript i Baseline equals left-parenthesis ModifyingAbove u With caret Subscript i Baseline 1 Baseline comma ellipsis comma ModifyingAbove u With caret Subscript i m Baseline right-parenthesis, i equals 1 comma ellipsis comma n. One commonly used transformation is the nonparametric estimation of the CDF of the marginal distributions, which is closely related to empirical CDF,

ModifyingAbove u With caret Subscript i comma j Baseline equals ModifyingAbove upper F With caret Subscript j comma n Baseline left-parenthesis x Subscript i comma j Baseline right-parenthesis

where

ModifyingAbove upper F With caret Subscript j comma n Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over n plus 1 EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts upper I Subscript left-bracket x Sub Subscript i comma j Subscript less-than-or-equal-to x right-bracket

The transformed data ModifyingAbove u With caret Subscript i comma j are used as if they had uniform marginal distributions; hence, they are called pseudo-samples. The function ModifyingAbove upper F With caret Subscript j comma n is different from the standard empirical CDF in the scalar 1 slash left-parenthesis n plus 1 right-parenthesis, which is to ensure that the transformed data cannot be on the boundary of the unit interval left-bracket 0 comma 1 right-bracket. It is clear that

ModifyingAbove u With caret Subscript i comma j Baseline equals StartFraction 1 Over n plus 1 EndFraction rank left-parenthesis x Subscript i comma j Baseline right-parenthesis

where rank left-parenthesis x Subscript i comma j Baseline right-parenthesis is the rank among i equals 1 comma ellipsis comma n in increasing order.

Let c left-parenthesis u 1 comma u 2 comma ellipsis comma u Subscript m Baseline semicolon theta right-parenthesis be the density function of a copula upper C left-parenthesis u 1 comma u 2 comma ellipsis comma u Subscript m Baseline semicolon theta right-parenthesis, and let theta be the parameter vector to be estimated. The parameter theta is estimated by maximum likelihood:

ModifyingAbove theta With caret equals arg max Underscript theta element-of normal upper Theta Endscripts sigma-summation Underscript i equals 1 Overscript n Endscripts log c left-parenthesis ModifyingAbove u With caret Subscript i Baseline 1 Baseline comma ellipsis comma ModifyingAbove u With caret Subscript i m Baseline semicolon theta right-parenthesis
Last updated: June 19, 2025