# acR: Distribution of the Radial Part of an Archimedean Copula In copula: Multivariate Dependence with Copulas

 acR R Documentation

## 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 June 15, 2022, 5:07 p.m.