copula (version 1.1-4)

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))

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.

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.

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.

See Also

evCopula, A, and evTestA. Further, evTestC, evTestK, exchEVTest, and gofEVCopula.

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))