copBasic (version 2.1.5)

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

Description

The N4212 copula (Nelsen, 2006, p. 91; eq. 4.2.12) is named by the author (Asquith) for the copBasic package and is defined as $$\mathbf{C}_{\mathrm{N4212}}(u,v; \Theta) = \biggl(1 + \bigl[(u^{-1} -1)^\Theta + (v^{-1} -1)^\Theta\bigr]^{1/\Theta}\biggr)^{-1}\mbox{.}$$

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 to test against.

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 infis = \(\Theta > 100\) 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

# NOT RUN {
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
# }
# NOT RUN {
D <- simCOP(n=400, cop=N4212cop, para=2)
D <- simCOP(n=400, cop=N4212cop, para=10,  ploton=FALSE, col=2)
D <- simCOP(n=400, cop=N4212cop, para=100, ploton=FALSE, col=3)#
# }