copBasic (version 2.0.1)

N4212cop: The Copula of Equation 4.2.12 of Nelsen's Book

Description

The N4212 copula (Nelsen, 2006, p. 91) is named by the author for the copBasic package and is defined as $${\mathbf{C}}_{\mathrm{N}4212}(u,v;\mathrm{\Theta })=(1+[(u^{-1}-1{)}^{\mathrm{\Theta }}+(v^{-1}-1{)}^{\mathrm{\Theta }}{]}^{1/\mathrm{\Theta }}{)}^{-1}\text{.}$$

The $\mathbf{N4212}(u,v)$ copula is not comprehensive because for $\Theta = 1$ the copula becomes the so-called $\mathbf{PSP}(u,v)$ copula (see PSP) and as $\Theta \rightarrow \infty$ the copula becomes $\mathbf{M}(u,v)$ (see M). The copula is undefined for $\Theta < 1$. The N4212 copula has respective lower- and upper-tail dependency (see taildepCOP).

Although copBasic is intended to not implement or store house the enormous suite of copula functions available in the literature, the N4212 copula is included to give the package another copula with a parameter besides the Gumbel-Hougaard (GHcop), Plackett (PLACKETTcop), and a few other copulas. This is done largely for pedagogic purposes. But the Plackett is comprehensive and N4212 is not comprehensive so both those concepts are available for instructional purposes.

Usage

N4212cop(u, v, para=NULL, infis=100, ...)

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;

infis

What is infinity? Testing shows that $\Theta > 100$ is about right to consider the copula as becoming $\mathbf{M}(u,v)$ (see M); and

...

Additional arguments to pass.

Value

• Value(s) for the copula are returned.

References

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

Examples

N4212cop(0.4,0.6, para=1) == PSP(0.4,0.6) # TRUE
N4212cop(0.4,0.6, para=10) # 0.3999928
taildepCOP(cop=N4212cop, para=10) # LamL = 0.93303; LamU = 0.92823
D <- simCOP(n=400, cop=N4212cop, para=2)
D <- simCOP(n=400, cop=N4212cop, para=10)
D <- simCOP(n=400, cop=N4212cop, para=100)