copula (version 0.999-18)

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. Ne<U+0161>lehov<U+00E1> (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. Ne<U+0161>lehov<U+00E1>, J. (2009). Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions. The Annals of Statistics 37(5b), 3059--3097.

Examples

# NOT RUN {
## 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])
# }