Construction of copulas using Khoudraji's device
Creates an object representing a copula constructed using
Khoudraji's device (Khoudraji, 1995). The resulting R object
is either of class "khoudrajiBivCopula",
"khoudrajiExplicitCopula"
or "khoudrajiCopula".
In the bivariate case, given two copulas C_1 and C_2, Khoudraji's device consists of defining a copula whose c.d.f. is given by:
C_1(u_1^{1 - a_1}, u_2^{1 - a_2}) C_2(u_1^{a_1}, u_2^{a_2})
where a_1 and a_2 are shape parameters in [0,1].
The construction principle (see also Genest et al. 1998) is a special case of that considered in Liebscher (2008).
khoudrajiCopula(copula1 = indepCopula(), copula2 = indepCopula(dim = d),
shapes = rep(NA_real_, dim(copula1)))If the argument copulas are bivariate, an object of class
"khoudrajiBivCopula" will be constructed. If
they are exchangeable and d-dimensional with d > 2, and if
they have explicit p.d.f. and c.d.f. expressions, an object of class
"khoudrajiExplicitCopula" will be
constructed. For the latter two classes, density evaluation is implemented,
and fitting and goodness-of-fit testing can be attempted. If d >
2 but one of the argument copulas does not have explicit p.d.f. and
c.d.f. expressions, or is not exchangeable, an object of class
"khoudrajiCopula" will be constructed, for which
density evaluation is not possible.
A new object of class "khoudrajiBivCopula" in
dimension two or of class
"khoudrajiExplicitCopula" or
"khoudrajiCopula" when d > 2.
Genest, C., Ghoudi, K., and Rivest, L.-P. (1998), Discussion of "Understanding relationships using copulas", by Frees, E., and Valdez, E., North American Actuarial Journal 3, 143–149.
Khoudraji, A. (1995), Contributions à l'étude des copules et àla modélisation des valeurs extrêmes bivariées, PhD thesis, Université Laval, Québec, Canada.
Liebscher, E. (2008), Construction of asymmetric multivariate copulas, Journal of Multivariate Analysis 99, 2234–2250.
## A bivariate Khoudraji-Clayton copula
kc <- khoudrajiCopula(copula2 = claytonCopula(6),
shapes = c(0.4, 0.95))
class(kc) # "kh..._Biv_Copula"
kc
contour(kc, dCopula, nlevels = 20, main = "dCopula(<khoudrajiBivCopula>)")
## A Khoudraji-Clayton copula with second shape parameter fixed
kcf <- khoudrajiCopula(copula2 = claytonCopula(6),
shapes = fixParam(c(0.4, 0.95), c(FALSE, TRUE)))
kcf. <- setTheta(kcf, c(3, 0.2)) # (change *free* param's only)
validObject(kcf) & validObject(kcf.)
## A "nested" Khoudraji bivariate copula
kgkcf <- khoudrajiCopula(copula1 = gumbelCopula(3),
copula2 = kcf,
shapes = c(0.7, 0.25))
kgkcf # -> 6 parameters (1 of 6 is 'fixed')
contour(kgkcf, dCopula, nlevels = 20,
main = "dCopula(<khoudrajiBivC.(nested)>)")
(Xtras <- copula:::doExtras()) # determine whether examples will be extra (long)
n <- if(Xtras) 300 else 64 # sample size (realistic vs short for example)
u <- rCopula(n, kc)
plot(u)
## For likelihood (or fitting), specify the "free" (non-fixed) param's:
## C1: C2c C2s1 sh1 sh2
loglikCopula(c(3, 6, 0.4, 0.7, 0.25),
u = u, copula = kgkcf)
## Fitting takes time (using numerical differentiation) and may be difficult:
## Starting values are required for all parameters
f.IC <- fitCopula(khoudrajiCopula(copula2 = claytonCopula()),
start = c(1.1, 0.5, 0.5), data = pobs(u),
optim.method = "Nelder-Mead")
summary(f.IC)
confint(f.IC) # (only interesting for reasonable sample size)
## Because of time, don't run these by default :
## Second shape parameter fixed to 0.95
kcf2 <- khoudrajiCopula(copula2 = claytonCopula(),
shapes = fixParam(c(NA_real_, 0.95), c(FALSE, TRUE)))
system.time(
f.ICf <- fitCopula(kcf2, start = c(1.1, 0.5), data = pobs(u),
optim.method = "Nelder-Mead")
) # ~ 7-8 sec
confint(f.ICf) # !
coef(f.ICf, SE=TRUE)
## With a different optimization method
system.time(
f.IC2 <- fitCopula(kcf2, start = c(1.1, 0.5), data = pobs(u),
optim.method = "BFGS")
)
printCoefmat(coef(f.IC2, SE=TRUE), digits = 3) # w/o unuseful extra digits
if(Xtras >= 2) { # really S..L..O..W... --------
## GOF example
optim.method <- "Nelder-Mead" #try "BFGS" as well
gofCopula(kcf2, x = u, start = c(1.1, 0.5), optim.method = optim.method)
gofCopula(kcf2, x = u, start = c(1.1, 0.5), optim.method = optim.method,
sim = "mult")
## The goodness-of-fit tests should hold their level
## but this would need to be tested
## Another example under the alternative
u <- rCopula(n, gumbelCopula(4))
gofCopula(kcf2, x = u, start = c(1.1, 0.5), optim.method = optim.method)
gofCopula(kcf2, x = u, start = c(1.1, 0.5), optim.method = optim.method,
sim = "mult")
}## ------ end { really slow gofC*() } --------
## Higher-dimensional constructions
## A three dimensional Khoudraji-Clayton copula
kcd3 <- khoudrajiCopula(copula1 = indepCopula(dim=3),
copula2 = claytonCopula(6, dim=3),
shapes = c(0.4, 0.95, 0.95))
n <- if(Xtras) 1000 else 100 # sample size (realistic vs short for example)
u <- rCopula(n, kcd3)
splom2(u)
v <- matrix(runif(15), 5, 3)
dCopula(v, kcd3)
## A four dimensional Khoudraji-Normal copula
knd4 <- khoudrajiCopula(copula1 = indepCopula(dim=4),
copula2 = normalCopula(.9, dim=4),
shapes = c(0.4, 0.95, 0.95, 0.95))
knd4
stopifnot(class(knd4) == "khoudrajiCopula")
u <- rCopula(n, knd4)
splom2(u)
## TODO :
## dCopula(v, knd4) ## not implementedPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.