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

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

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 or test in numerical examples.

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.

Author(s)

W.H. Asquith

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
## 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)#
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.