Exponent measure for multivariate generalized Pareto distributions

Integrated intensity over the region defined by \([0, z]^c\) for logistic, Huesler-Reiss, Brown-Resnick and extremal Student processes.

Usage

expme(
  z,
  par,
  model = c("log", "hr", "br", "xstud"),
  method = c("TruncatedNormal", "mvtnorm", "mvPot")
)

Arguments

z

vector at which to estimate exponent measure

par

list of parameters

model

string indicating the model family

method

string indicating the package from which to extract the numerical integration routine

Value

numeric giving the measure of the complement of \([0,z]\).

Note

The list par must contain different arguments depending on the model. For the Brown--Resnick model, the user must supply the conditionally negative definite matrix Lambda following the parametrization in Engelke et al. (2015) or the covariance matrix Sigma, following Wadsworth and Tawn (2014). For the Husler--Reiss model, the user provides the mean and covariance matrix, m and Sigma. For the extremal student, the covariance matrix Sigma and the degrees of freedom df. For the logistic model, the strictly positive dependence parameter alpha.

Examples

if (FALSE) {
# Extremal Student
Sigma <- stats::rWishart(n = 1, df = 20, Sigma = diag(10))[, , 1]
expme(z = rep(1, ncol(Sigma)), par = list(Sigma = cov2cor(Sigma), df = 3), model = "xstud")
# Brown-Resnick model
D <- 5L
loc <- cbind(runif(D), runif(D))
di <- as.matrix(dist(rbind(c(0, ncol(loc)), loc)))
semivario <- function(d, alpha = 1.5, lambda = 1) {
  (d / lambda)^alpha
}
Vmat <- semivario(di)
Lambda <- Vmat[-1, -1] / 2
expme(z = rep(1, ncol(Lambda)), par = list(Lambda = Lambda), model = "br", method = "mvPot")
Sigma <- outer(Vmat[-1, 1], Vmat[1, -1], "+") - Vmat[-1, -1]
expme(z = rep(1, ncol(Lambda)), par = list(Lambda = Lambda), model = "br", method = "mvPot")
}