PARETOcop: The Pareto Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

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 the Nelsen (2006) presentation because it is anticipated that copBasic users are more likely to have right-tail dependency situations (say large maxima [right tail] coupling in earth-system data but not small maxima [left tail] coupling).

Usage

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

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; and
...	Additional arguments to pass.

Value

Value(s) for the copula are returned.

Note

The Pareto copula is used in a demonstration of Kendall Function L-moment ratio diagram construction (see kfuncCOPlmoms).

Author(s)

W.H. Asquith

References

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

See Also

M, P

Examples

## 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."
## End(Not run)

copBasic documentation built on Oct. 17, 2023, 5:08 p.m.