An {copula} | R Documentation

## 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)
    An(x, 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.

### 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. arXiv:1107.2410v1.

### 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))

[Package copula version 0.999-15 Index]