PLACKETTcop {copBasic} | R Documentation |
The Plackett Copula
Description
The Plackett copula (Nelsen, 2006, pp. 89–92) is
The Plackett copula () is comprehensive because as
the copula becomes
(see
W
, countermonotonicity), as the copula becomes
(see
M
, comonotonicity) and for the copula is
(see
P
, independence).
Nelsen (2006, p. 90) shows that
where and
are cumulative distribution function for random variables
and
, respectively, and
is the joint distribution function. Only Plackett copulas have a constant
for any pair
. 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
contingency table. Consider the
contingency table shown at the end of this section, then
is defined as
where it is obvious that and
,
,
, and
can be replaced by proporations for a sample of size
by
,
,
, and
, 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
and
(
taildepCOP
).
| Low | High | Sums |
Low | | |
|
High | | |
|
Sums | | |
|
Usage
PLACKETTcop(u, v, para=NULL, ...)
PLcop(u, v, para=NULL, ...)
Arguments
u |
Nonexceedance probability |
v |
Nonexceedance probability |
para |
A vector (single element) of parameters—the |
... |
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)