acR: Distribution of the Radial Part of an Archimedean Copula

In copula: Multivariate Dependence with Copulas

Distribution of the Radial Part of an Archimedean Copula

Description

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.

Usage

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, ...)

Arguments

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	logical; if TRUE, probabilities are P[X <= x] otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are given as log(p).
interval	root-search interval.
tol	see uniroot().
maxiter	see uniroot().
...	additional arguments passed to the procedure for computing derivatives.

Value

The distribution function of the radial part evaluated at x, or its inverse, the quantile at p.

References

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.

Examples

## 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])

copula documentation built on Feb. 16, 2023, 8:46 p.m.