copula (version 0.999-7)

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{Neslehova} (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, ...)

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.

...

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{Neslehova}, 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 = expression(italic(x)),
     ylab = substitute(italic(F[R](x))~~"for d=1:"*dm, list(dm=dmax)))
for(k in 2:dmax) lines(x, y[,k])