copBasic (version 1.7.1)

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 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).

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 Plackett 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

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

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, the Plackett 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 $\lambda_U = 0$ (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

Joe, H., 2015, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

See Also

PLACKETTpar, PLACKETTsim, W, M, densityCOP

Examples

PLACKETTcop(0.4,0.6,para=1)
P(0.4,0.6) # same two values because Theta == 1 is Independence
PLACKETTcop(0.4,0.6,para=10.25)

# Joe (2015, p. 164) shows the closed form copula density of the Plackett.
"dPLACKETTcop" <- function(u,v,para) {
   eta <- para - 1; A <- para*(1 + eta*(u+v-2*u*v))
   B <- ((1 + eta*(u+v))^2 - 4*para*eta*u*v)^(3/2); return(A/B)
}
u <- 0.08; v <- 0.67 # Two probabilities to make numerical evaluations.
del <- 0.0001 # a 'small' differential value of probability
u1 <- u; u2 <- u+del; v1 <- v; v2 <- v+del
# Density following (Nelsen, 2006, p. 10)
dCrect <- (PLACKETTcop(u2, v2, para=10.25) - PLACKETTcop(u2, v1, para=10.25) -
           PLACKETTcop(u1, v2, para=10.25) + PLACKETTcop(u1, v1, para=10.25))/del^2
dCanal <- dPLACKETTcop(u,  v,  para=10.25)
dCfunc <- densityCOP(u, v, cop=PLACKETTcop, para=10.25, deluv = del)
R <- round(c(dCrect, dCanal, dCfunc), digits=6)
message("Density: ", R[1], "(manual), ", R[2], "(analytical), ", R[3], "(function)");

# Comparison of partial derivatives
dUr <- (PLACKETTcop(u2, v2, para=10.25) - PLACKETTcop(u1, v2, para=10.25))/del
dVr <- (PLACKETTcop(u2, v2, para=10.25) - PLACKETTcop(u2, v1, para=10.25))/del
dU  <-  derCOP(u, v, cop=PLACKETTcop, para=10.25)
dV  <- derCOP2(u, v, cop=PLACKETTcop, para=10.25)
R   <- round(c(dU, dV, dUr, dVr), digits=6)
message("Partial derivatives dU=", R[1], "and dUr=", R[3], "",
        "dV=", R[2], "and dVr=", R[4])