onacopula: Constructing (Outer) Nested Archimedean Copulas

Description

Constructing (outer) nested Archimedean copulas (class outer_nacopula) is most conveniently done via onacopula(), using a nested $C(...)$ notation.

Slightly less conveniently, but with the option to pass a list structure, onacopulaL() can be used, typically from inside another function programmatically.

Usage

onacopula (family, nacStructure)
onacopulaL(family, nacList)
nac2list(x)

Arguments

family

either a character string, the short or longer form of the Archimedean family name (for example, "Clayton" or simply "C"); see the acopula-families documentation, or an acopula family object.

nacStructure

a “formula” of the form

C (θ, c (i_{1}, \dots, i_{c}), list (C (. .), . . ., C (. .))) .

Note that $C()$ has (maximally) three arguments: the first is the copula parameter (vector) $theta$, the second a (possibly empty) vector of integer indices of components (for the comp slot in nacopulas), and finally a (possibly empty) list of child copulas, each specified with in the $C(..)$ notation themselves.

nacList

a list of length 3 (or 2), with elements

theta: $\theta$
comp: components $c(i1,..ic)$
children: a list which must be a nacList itself and may be missing to denote the empty list().

an "nacopula", (typically "outer_nacopula") object.

Value

onacopula[L](): An outer nested Archimedean copula object, that is, of class "outer_nacopula".nac2list: a list exactly like the naclist argument to onacopulaL.

References

Those of the Archimedean families, for example, copGumbel.

Examples

Run this code

## Construct a ten-dimensional Joe copula with parameter such that
## Kendall's tau equals 0.5
theta <- copJoe@iTau(0.5)
C10 <- onacopula("J",C(theta,1:10))

## Equivalent construction with onacopulaL():
C10. <- onacopulaL("J",list(theta,1:10))
stopifnot(identical(C10, C10.),
          identical(nac2list(C10), list(theta, 1:10)))

## Construct a three-dimensional nested Gumbel copula with parameters
## such that Kendall's tau of the respective bivariate margins are 0.2
## and 0.5.
theta0 <- copGumbel@iTau(.2)
theta1 <- copGumbel@iTau(.5)
C3 <- onacopula("G", C(theta0, 1, C(theta1, c(2,3))))

## Equivalent construction with onacopulaL():
str(NAlis <- list(theta0, 1, list(list(theta1, c(2,3)))))
C3. <- onacopulaL("Gumbel", NAlis)
stopifnot(identical(C3, C3.))

## An exercise: assume you got the copula specs as character string:
na3spec <- "C(theta0, 1, C(theta1, c(2,3)))"
na3call <- parse(text = na3spec)[[1]]
C3.s <- onacopula("Gumbel", na3call)
stopifnot(identical(C3, C3.s))

## Good error message if the component ("coordinate") indices are wrong
## or do not match:
err <- try(onacopula("G", C(theta0, 2, C(theta1, c(3,2)))))

## Compute the probability of falling in [0,.01]^3 for this copula
pCopula(rep(.01,3), C3)

## Compute the probability of falling in the cube [.99,1]^3
prob(C3, rep(.99, 3), rep(1, 3))

## Construct a 6-dimensional, partially nested Gumbel copula of the form
## C_0(C_1(u_1, u_2), C_2(u_3, u_4), C_3(u_5, u_6))
theta <- 2:5
copG <- onacopulaL("Gumbel", list(theta[1], NULL, list(list(theta[2], c(1,2)),
                                                       list(theta[3], c(3,4)),
                                                       list(theta[4], c(5,6)))))
set.seed(1)
U <- rCopula(5000, copG)
pairs(U, pch=".", gap=0, labels=as.expression( sapply(1:dim(copG),
    function(j) bquote(italic(U[.(j)]))) ))

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