PLACKETTcop: The Plackett Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

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, ...)

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 Plackett copula was the first (2008) copula implemented in copBasic as part of initial development of the code base for instructional purposes. Thus, this particular copula has a separate parameter estimation function in PLACKETTpar as a historical vestige of a class project.

Author(s)

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.

See Also

PLACKETTpar, PLpar, PLACKETTsim, W, M, densityCOP

Examples

PLACKETTcop(0.4, 0.6, para=1)
P(0.4, 0.6) # independence copula, same two values because Theta == 1
PLcop(0.4, 0.6, para=10.25) # joint probability through positive association

## Not run: 
# Joe (2014, 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 <- (PLcop(u2, v2, para=10.25) - PLcop(u2, v1, para=10.25) -
           PLcop(u1, v2, para=10.25) + PLcop(u1, v1, para=10.25)) / del^2
dCanal <- dPLACKETTcop(u, v, para=10.25)
dCfunc <-   densityCOP(u, v, para=10.25, cop=PLcop, deluv = del)
R <- round(c(dCrect, dCanal, dCfunc), digits=6)
message("Density: ", R[1], "(manual), ", R[2], "(analytical), ", R[3], "(function)");
# Density: 0.255377(manual), 0.255373(analytical), 0.255377(function)

# Comparison of partial derivatives
dUr <- (PLcop(u2, v2, para=10.25) - PLcop(u1, v2, para=10.25)) / del
dVr <- (PLcop(u2, v2, para=10.25) - PLcop(u2, v1, para=10.25)) / del
dU  <- derCOP( u, v, cop=PLcop, para=10.25)
dV  <- derCOP2(u, v, cop=PLcop, para=10.25)
R   <- round(c(dU, dV, dUr, dVr), digits=6)
message("Partial derivatives dU=", R[1], " and dUr=", R[3], "\n",
        "                    dV=", R[2], " and dVr=", R[4]) #
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.