The Plackett copula (Nelson, 2006, pp. 89--92) is
$$\mathbf{C}_{\Theta}(u,v) = \frac{[1+(\Theta-1)(u+v)]-\sqrt{[1+(\Theta-1)(u+v)]^2 - 4uv\Theta(\Theta-1)}}{2(\Theta - 1)}\mbox{.}$$
The Plackett copula is comprehensive because as $\Theta \rightarrow 0$ the copula becomes $\mathbf{W}(u,v)$ (see W), as $\Theta \rightarrow \infty$ the copula becomes $\mathbf{M}(u,v)$ (see M) and for $\Theta = 1$ the copula is $\mathbf{\Pi}(u,v)$ (see P, independence). The Plackett copula has been widely used in modeling and as an alternative to bivariate distributions. The Plackett copula has respective lower- and upper-tail dependency parameters of $\lambda_L = 0$ and $\lambda_U = 0$ (see taildepCOP).
Usage
PLACKETTcop(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.
concept
comprehensive copula
References
Nelson, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.