acR {copula}R Documentation

Distribution of the Radial Part of an Archimedean Copula

Description

pacR() computes the distribution function FRF_R of the radial part of an Archimedean copula, given by

FR(x)=1k=0d1(x)kψ(k)(x)k!, x[0,);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 FRF_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 FRF_R.

p

numeric vector of evaluation points of the quantile function.

family

Archimedean family.

theta

parameter thetatheta.

d

dimension dd.

lower.tail

logical; if TRUE, probabilities are P[X<=x]P[X <= x] otherwise, P[X>x]P[X > x].

log.p

logical; if TRUE, probabilities pp are given as logp\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, dd-monotone functions and l1l_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])

[Package copula version 1.1-3 Index]