N4220cop: The Copula of Equation 4.2.20 of Nelsen's Book

In copBasic: General Bivariate Copula Theory and Many Utility Functions

The Copula of Equation 4.2.20 of Nelsen's Book

Description

The N4220 copula (Nelsen, 2006, p. 118; eq. 4.2.20) is named by the author (Asquith) for the copBasic package and is defined as

\mathbf{C}_{\mathrm{N4220}}(u,v; \Theta) = \biggl[\mathrm{log}\biggl( \mathrm{exp}\bigl(u^{-\Theta}\bigr) + \mathrm{exp}\bigl(v^{-\Theta}\bigr) - \mathrm{exp}(1)\biggr)\biggr]^{-1/\Theta}\mbox{.}

The \mathbf{N4220}(u,v) copula is not comprehensive because for \Theta = 0^{+} the copula becomes the independence \mathbf{P}(u,v) copula (P) and as \Theta \rightarrow \infty the copula becomes \mathbf{M}(u,v) (M). The copula is undefined for \Theta < 0. This copula has lower-tail dependency, which is the reverse of the \mathbf{rN4220}(u,v) that has uppr-tail dependency. Hence, a reflected (refer to COP, reflect=2) version of this copula is natively supported as the rN4220cop to put the tail dependency into the upper-right corner and more “hydrologic-system like.”

Usage

N4220cop(u, v, para=NULL, insertM=TRUE, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
para	A vector (single element) of parameters—the \Theta parameter of the copula;
insertM	For the tail that pinches out towards \mathbf{M}(u,v) (M) use an ad hoc algorithm to insert that copula into the tail (refer to source code) and that algorithm is dependent on the simCOP and the numerical partial derivative inversion for simulation that the package is based around; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Examples

sapply(c(0.1, 1, 10), function(p) N4220cop(0.4, 0.6, para=p))
# 0.2611771 0.3714941 0.4000000 # perfect correlation controls for p=10

copBasic documentation built on July 23, 2026, 1:07 a.m.