Distribution of the Radial Part of an Archimedean Copula
pacR() computes the distribution function F_R of the radial
part of an Archimedean copula, given by
F_R(x)=1-sum(k=0,...,d-1) (-x)^k psi^{(k)}(x)/k!, u in [0,Inf)
The formula (in a slightly more general form) is given by McNeil and G. Nešlehová (2009).
qacR() computes the quantile function of F_R.
pacR(x, family, theta, d, lower.tail = TRUE, log.p = FALSE, ...)
qacR(p, family, theta, d, log.p = FALSE, interval,
tol = .Machine$double.eps^0.25, maxiter = 1000, ...)x |
numeric vector of nonnegative evaluation points for F_R. |
p |
numeric vector of evaluation points of the quantile function. |
family |
Archimedean family. |
theta |
parameter theta. |
d |
dimension d. |
lower.tail |
|
log.p |
|
interval |
root-search interval. |
tol |
see |
maxiter |
see |
... |
additional arguments passed to the procedure for computing derivatives. |
The distribution function of the radial part evaluated at
x, or its inverse, the quantile at p.
McNeil, A. J., G. Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l_1-norm symmetric distributions. The Annals of Statistics 37(5b), 3059–3097.
## setup
family <- "Gumbel"
tau <- 0.5
m <- 256
dmax <- 20
x <- seq(0, 20, length.out=m)
## compute and plot pacR() for various d's
y <- vapply(1:dmax, function(d)
pacR(x, family=family, theta=iTau(archmCopula(family), tau), d=d),
rep(NA_real_, m))
plot(x, y[,1], type="l", ylim=c(0,1),
xlab = quote(italic(x)), ylab = quote(F[R](x)),
main = substitute(italic(F[R](x))~~ "for" ~ d==1:.D, list(.D = dmax)))
for(k in 2:dmax) lines(x, y[,k])Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.