PARETOcop: The Pareto Copula

Description

The Pareto copula (Nelsen, 2006, pp. 33) is $$\mathbf{C}_{\Theta}(u,v) = \mathbf{PA}(u,v) = \bigl[(1-u)^{-\Theta}+(1-v)^{-\Theta}\bigr]^{-1/\Theta}\mbox{,}$$ where $\Theta \in [0, \infty)$. As $\Theta \rightarrow 0^{+}$, the copula limits to the $\mathbf{\Pi}$ copula (P) and the $\mathbf{M}$ copula (M). The parameterization here has assocation increasing with increasing $\Theta$, which differs from Nelsen (2006), and also the Pareto copula is formed with right-tail increasing reflection of Nelsen's presentation because it is anticipated that copBasic users are more likely to have right-tail dependency situations (say large maxima [right tail] coupling but not small maxima [left tail] coupling).

Usage

PARETOcop(u, v, para=NULL, ...)
    PAcop(u, v, para=NULL, ...)

Arguments

Nonexceedance probability $u$ in the $X$ direction;

Nonexceedance probability $v$ in the $Y$ direction;

para

A vector (single element) of parameters---the $\Theta$ parameter of the copula; 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

Run this code

# NOT RUN {
z <- seq(0.01,0.99, by=0.01) # Both copulas have Kendall Tau = 1/3
plot( z, kfuncCOP(z, cop=PAcop, para=1), lwd=2,
                                xlab="z <= Z", ylab="F_K(z)", type="l")
lines(z, kfuncCOP(z, cop=GHcop, para=1.5), lwd=2, col=2) # red line
# All extreme value copulas have the same Kendall Function [F_K(z)], the
# Gumbel-Hougaard is such a copula and the F_K(z) for the Pareto does not
# plot on top and thus is not an extreme value but shares a "closeness."
# }