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An

Nonparametric Rank-based Estimators of the Pickands Dependence Function

Description

Bivariate and multivariate versions of the nonparametric rank-based estimators of the Pickands dependence function A, studied in Genest and Segers (2009) and Gudendorf and Segers (2011).

Usage

An.biv(x, w, estimator = c("CFG", "Pickands"), corrected = TRUE,
       ties.method = eval(formals(rank)$ties.method))
An(x, w, ties.method = eval(formals(rank)$ties.method))

Arguments

`x`	a data matrix that will be transformed to pseudo-observations. If `An.biv` is called, `x` has to have two columns.
`w`	if `An.biv` is called, a vector of points in [0,1] where to evaluate the estimated bivariate Pickands dependence function. If the multivariate estimator `An` is used instead, `w` needs to be a matrix with the same number of columns as `x` whose lines are elements of the multivariate unit simplex (see the last reference).
`estimator`	specifies which nonparametric rank-based estimator of the unknown Pickands dependence function to use in the bivariate case; can be either `"CFG"`(Capéraà-Fougères-Genest) or `"Pickands"`.
`corrected`	TRUE means that the bivariate estimators will be corrected to ensure that their value at 0 and 1 is 1.
`ties.method`	`character` string specifying how ranks should be computed if there are ties in any of the coordinate samples of `x`; passed to `pobs`.

Details

More details can be found in the references.

Value

An.biv() returns a vector containing the values of the estimated Pickands dependence function at the points in w (and is the same as former Anfun()).

The function An computes simultaneously the three corrected multivariate estimators studied in Gudendorf and Segers (2011) at the points in w and retuns a list whose components are

`P`	values of the Pickands estimator at the points in `w`.
`CFG`	values of the CFG estimator at the points in `w`.
`HT`	values of the Hall-Tajvidi estimator at the points in `w`.

References

C. Genest and J. Segers (2009). Rank-based inference for bivariate extreme-value copulas. Annals of Statistics 37, 2990–3022.

G. Gudendorf and J. Segers (2011). Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference 142, 3073–3085.

Examples

## True Pickands dependence functions
curve(A(gumbelCopula(4   ), x), 0, 1)
curve(A(gumbelCopula(2   ), x), add=TRUE, col=2)
curve(A(gumbelCopula(1.33), x), add=TRUE, col=3)

## CFG estimator
curve(An.biv(rCopula(1000, gumbelCopula(4   )), x), lty=2, add=TRUE)
curve(An.biv(rCopula(1000, gumbelCopula(2   )), x), lty=2, add=TRUE, col=2)
curve(An.biv(rCopula(1000, gumbelCopula(1.33)), x), lty=2, add=TRUE, col=3)

## Pickands estimator
curve(An.biv(rCopula(1000, gumbelCopula(4   )), x, estimator="Pickands"),
      lty=3, add=TRUE)
curve(An.biv(rCopula(1000, gumbelCopula(2   )), x, estimator="Pickands"),
      lty=3, add=TRUE, col=2)
curve(An.biv(rCopula(1000, gumbelCopula(1.33)), x, estimator="Pickands"),
      lty=3, add=TRUE, col=3)

legend("bottomleft",  paste0("Gumbel(", format(c(4, 2, 1.33)),")"),
       lwd=1, col=1:3, bty="n")
legend("bottomright", c("true", "CFG est.", "Pickands est."), lty=1:3, bty="n")

## Relationship between An.biv and An
u <- c(runif(100),0,1) # include 0 and 1
x <- rCopula(1000, gumbelCopula(4))
r <- An(x, cbind(1-u, u))
all.equal(r$CFG, An.biv(x, u))
all.equal(r$P, An.biv(x, u, estimator="Pickands"))

## A trivariate example
x <- rCopula(1000, gumbelCopula(4, dim = 3))
u <- matrix(runif(300), 100, 3)
w <- u / apply(u, 1, sum)
r <- An(x, w)

## Endpoint corrections are applied
An(x, cbind(1, 0, 0))
An(x, cbind(0, 1, 0))
An(x, cbind(0, 0, 1))

copula

Multivariate Dependence with Copulas

v1.0-1

GPL (>= 3) | file LICENCE

Authors

Marius Hofert [aut] (<https://orcid.org/0000-0001-8009-4665>), Ivan Kojadinovic [aut] (<https://orcid.org/0000-0002-2903-1543>), Martin Maechler [aut, cre] (<https://orcid.org/0000-0002-8685-9910>), Jun Yan [aut] (<https://orcid.org/0000-0003-4401-7296>), Johanna G. Nešlehová [ctb] (evTestK(), <https://orcid.org/0000-0001-9634-4796>), Rebecca Morger [ctb] (fitCopula.ml(): code for free mixCopula weight parameters)

Initial release

2020-12-07

An