PickandsDF {AoE}R Documentation

Pickands Dependence Function

Description

Computes an estimate of the Pickands dependence function of the extreme-value attractor of a bivariate distribution based on a bivariate sample (X_1, Y_1), ..., (X_n, Y_n) from that distribution.

Usage

PickandsDF(data.x, data.y, data = NULL, w = (0:20)/20, k, 
        method = "empirical", plot = TRUE)

Arguments

data.x, data.y Numeric vectors containing the data X_1, ..., X_n and Y_1, ..., Y_n, respectively.
data Alternatively, the data may be provided in the form of a n-by-2 matrix. If provided, then the arguments data.x and data.y are ignored.
w A numeric vector giving the points in which the estimate of the Pickands dependence function will be computed. All the elements should be between 0 and 1.
k A number specifying the tuning parameter k at which the estimate will be computed. See ‘Details’.
method A string specifying the estimation method; possible choices are "empirical" and "angular". See ‘Details’.
plot If TRUE (the default), the result will be plotted.

Details

The Pickands dependence function of a bivariate extreme-value distribution is defined by

A(w) = l(w, 1-w)

for 0 <= w <= 1, where l is the (stable) tail dependence function; see ETDF for more details. Conversely,

l(x, y) = (x+y) A(x / (x+y))

for x, y >= 0 and x + y > 0.

A function A from [0, 1] to [0, 1/2] is a Pickands dependence function if and only if (i) it is convex and (ii) max(w, 1-w) <= A(w) <= 1 for all 0 <= w <= 1. The lower bound corresponds to asymptotic complete dependence, the upper bound to asymptotic independence. The coefficient of tail dependence is given by lambda = 2 - l(1,1) = 2*(1 - A(0.5)).

If method = "empirical", the estimate is computed by

hat{A}(w) = hat{l}(w, 1-w)

where hat{l} is the empirical tail dependence function computed by ETDF at k = k. In general, the estimator does not satisfy any of the two requirements above.

If method = "angular", the estimate is computed by

hat{A}(w) = int_0^{π/2} max(w cos theta, (1-w) sin theta) dhat{Phi}(theta)

where hat{Phi} is an estimate of the angular measure; the estimator used is the estimator as implemented in AngularMeasure with k = k and method = "c". In this way, the estimator is guaranteed to satisfy the two constraints mentioned above. By exploiting the above relations between A and l, a nonparametric estimator of l is obtained that is itself a stable tail dependence function.

Value

A length(w)-by-length(methods) matrix containing at position (i,j) the point estimate at w[i] computed by method methods[j]. The result is returned invisibly.

References

Pickands, J. (1981) Multivariate extreme value distributions. In: Bulletin of the International Statistical Institute, Proceedings of the 43rd Session, Buenos Aires, pp. 859-878.

See Also

ETDF, AngularMeasure

Examples

x <- rbivcauchy(1000)
w <- seq(from = 0, to = 1, by = 0.02)
PickandsDF(data = x, w = w, k = 20, method = c("empirical", "angular"))
lines(w, sqrt(w^2 + (1-w)^2), col = "red")

[Package AoE version 1.0.1 Index]