PLACKETTcop: The Plackett Copula

Description

The Plackett copula (Nelsen, 2006, pp. 89--92) is $$\mathbf{C}_\Theta(u,v) = \mathbf{PL}(u,v) = \frac{[1+(\Theta-1)(u+v)]-\sqrt{[1+(\Theta-1)(u+v)]^2 - 4uv\Theta(\Theta-1)}}{2(\Theta - 1)}\mbox{.}$$

The Plackett copula ($\mathbf{PL}(u,v)$) is comprehensive because as $\Theta \rightarrow 0$ the copula becomes $\mathbf{W}(u,v)$ (see W, countermonotonicity), as $\Theta \rightarrow \infty$ the copula becomes $\mathbf{M}(u,v)$ (see M, comonotonicity) and for $\Theta = 1$ the copula is $\mathbf{\Pi}(u,v)$ (see P, independence).

Nelsen (2006, p. 90) shows that $$\Theta = \frac{H(x,y)[1 - F(x) - G(y) + H(x,y)]}{[F(x) - H(x,y)][G(y) - H(x,y)]}\mbox{,}$$ where $F(x)$ and $G(y)$ are cumulative distribution function for random variables $X$ and $Y$, respectively, and $H(x,y)$ is the joint distribution function. Only Plackett copulas have a constant $\Theta$ for any pair $\{x,y\}$. Hence, Plackett copulas are also known as constant global cross ratio or contingency-type distributions. The copula therefore is intimately tied to contingency tables and in particular the bivariate Plackett defined herein is tied to a $2\times2$ contingency table. Consider the $2\times 2$ contingency table shown at the end of this section, then $\Theta$ is defined as

$$\Theta = \frac{a/c}{b/d} = \frac{\frac{a}{a+c}/\frac{c}{a+c}}{\frac{b}{b+d}/\frac{d}{b+d}}\mbox{\ and\ }\Theta = \frac{a/b}{c/d} = \frac{\frac{a}{a+b}/\frac{b}{a+b}}{\frac{c}{c+d}/\frac{d}{c+d}}\mbox{,}$$

where it is obvious that $\Theta = ad/bc$ and $a$, $b$, $c$, and $d$ can be replaced by proporations for a sample of size $n$ by $a/n$, $b/n$, $c/n$, and $d/n$, respectively. Finally, this copula has been widely used in modeling and as an alternative to bivariate distributions and has respective lower- and upper-tail dependency parameters of $\lambda^L = 0$ and $\lambda^U = 0$ (taildepCOP).

${-}{-}$	Low	High	Sums
Low	$a$	$b$	$a+b$
High	$c$	$d$	$c+d$
Sums	$a+c$	$b+d$	${-}{-}$

Usage

PLACKETTcop(u, v, para=NULL, ...)
      PLcop(u, v, para=NULL, ...)

Value

Value(s) for the copula are returned.

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.

Author

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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