copula (version 1.1-3)

acR: 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 $$F_R(x)=1-\sum_{k=0}^{d-1} \frac{(-x)^k\psi^{(k)}(x)}{k!},\ x\in[0,\infty);$$ 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, ...)

Value

The distribution function of the radial part evaluated at

x, or its inverse, the quantile at p.

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.

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])