Simulation of generalized Huesler-Reiss Pareto vectors via composition sampling

Sample from the generalized Pareto process associated to Huesler-Reiss spectral profiles. For the Huesler-Reiss Pareto vectors, the matrix Sigma is utilized to build \(Q\) viz. $$Q = \Sigma^{-1} - \frac{\Sigma^{-1}\mathbf{1}_d\mathbf{1}_d^\top\Sigma^{-1}}{\mathbf{1}_d^\top\Sigma^{-1}\mathbf{1}_d}.$$ The location vector m and Sigma are the parameters of the underlying log-Gaussian process.

Usage

rparpcshr(n, u, alpha, Sigma, m)

Arguments

n

sample size

u

vector of marginal location parameters (must be strictly positive)

alpha

vector of shape parameters (must be strictly positive).

Sigma

covariance matrix of process, used to define \(Q\). See Details.

m

location vector of Gaussian distribution.

Value

n by d matrix of observations

References

Ho, Z. W. O and C. Dombry (2019), Simple models for multivariate regular variations and the Huesler-Reiss Pareto distribution, Journal of Multivariate Analysis (173), p. 525-550, doi:10.1016/j.jmva.2019.04.008

Examples

D <- 20L
coord <- cbind(runif(D), runif(D))
di <- as.matrix(dist(rbind(c(0, ncol(coord)), coord)))
semivario <- function(d, alpha = 1.5, lambda = 1){(d/lambda)^alpha}
Vmat <- semivario(di)
Sigma <- outer(Vmat[-1, 1], Vmat[1, -1], '+') - Vmat[-1, -1]
m <- Vmat[-1,1]
if (FALSE) {
samp <- rparpcshr(n = 100, u = c(rep(1, 10), rep(2, 10)),
          alpha = seq(0.1, 1, length = 20), Sigma = Sigma, m = m)
}