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).
PARETOcop(u, v, para=NULL, ...)
PAcop(u, v, para=NULL, ...)Value(s) for the copula are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A vector (single element) of parameters---the \(\Theta\) parameter of the copula; and
Additional arguments to pass.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
M, P