AngularMeasure {AoE}R Documentation

Angular Measure

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

AngularMeasure(data.x, data.y, data = NULL, k, 
        method = "u", 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.
k An numeric vector of values for k in the definition of the empirical tail dependence function; see ‘Details’.
method A character vector specifying the estimation method; possible choices are "u" for unconstrained and "c" for constrained. See ‘Details’.
plot If TRUE (the default), the estimated distribution functions will be plotted.

Details

This function is an implementation of the following nonparametric estimator for the angular or spectral measure Phi (de Haan and Resnick, 1977) of the extreme-value attractor of an unknown distribution. For data (X_1, Y_1), ..., (X_n, Y_n), let R_i be the rank of X_i among X_1, ..., X_n and let S_i be the rank of Y_i among Y_1, ..., Y_n. Define X_i^* = n / (n + 1 - R_i) and Y_i^* = n / (n + 1 - S_i). Write (X_i^*, Y_i^*) = (rho_i cos theta_i, rho_i sin theta_i) in polecoordinates. For 0 < k < n, let J be the set of i = 1, ..., n such that rho_i > n/k. Then the estimate hat{Phi} is the discrete measure with an atom of mass p_i at theta_i for all i in J. The masses or weights p_i depend on the method:

method = "u" unconstrained
Then p_i = 1/k for every i in J. This is called the empirical spectral measure and is a variant of the estimator considered for instance in Einmahl et al. (2001).
method = "c" constrained
Then the weights p_i are determined by a variant of maximum empirical likelihood taking into account the moment constraints that a spectral measure should satisfy (Einmahl and Segers 2007).

The argument k may be a vector, in which case, provided plot = TRUE, the corresponding distribution function hat{Phi}([0, theta]) will be drawn for every element of k. However, the value returned by the function corresponds only to the final element of k.

Value

A list with the class attribute "AngularMeasure", which is a list containing the following components:

angles The angles theta_i for i in J.
weights The corresponding weights p_i.
radii The full vector of radii rho_i for i = 1, ..., n.
indices The set J.

References

Einmahl, J.H.J., de Haan, L. and Piterbarg, V.I. (2001). Nonparametric estimation of the spectral measure of an extreme value distribution. The Annals of Statistics 29, 1401-1423.

Einmahl, J.H.J. and Segers, J. (2007). Maximum empirical likelihood estimation of the spectral measure of an extreme value distribution. In preparation.

de Haan, L. and Resnick, S.I. (1977). Limit theory for multivariate sample extremes. Zeitschrift fuer Wahrscheinlichkeitstheorie und Verwandte Gebiete 40, 317-337.

See Also

ETDF, PickandsDF

Examples

# For the bivariate Cauchy distribution on the positive quadrant,
# the angular measure is known to be Phi([0, theta]) = theta.
AngularMeasure(data = rbivcauchy(1000), k = c(20, 50), method = "c")
abline(a = 0, b = 1, col = "red")

[Package AoE version 1.0.1 Index]