AngularMeasure {AoE} | R Documentation |
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.
AngularMeasure(data.x, data.y, data = NULL, k, method = "u", plot = TRUE)
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. |
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"
unconstrainedmethod = "c"
constrained
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
.
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. |
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.
# 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")