tailDepFun (version 1.0.0)

EstimationBR: Estimation of the parameters of the Brown-Resnick process

Description

Estimation the parameters of the Brown-Resnick process, using either the pairwise M-estimator or weighted least squares (WLS).

Usage

EstimationBR(x, locations, indices, k, method, isotropic = FALSE, biascorr = FALSE, Tol = 1e-05, k1 = (nrow(x) - 10), tau = 5, startingValue = NULL, Omega = diag(nrow(indices)), iterate = FALSE, covMat = TRUE)

Arguments

x

An $n$ x $d$ data matrix.

locations

A $d$ x 2 matrix containing the Cartesian coordinates of $d$ points in the plane.

indices

A $q$ x $d$ matrix containing exactly 2 ones per row, representing a pair of points from the matrix locations, and zeroes elsewhere.

k

An integer between 1 and $n - 1$; the threshold parameter in the definition of the empirical stable tail dependence function.

method

Choose between Mestimator and WLS.

isotropic

A Boolean variable. If FALSE (the default), then an anisotropic process is estimated.

biascorr

For method = "WLS" only. If TRUE, then the bias-corrected estimator of the stable tail dependence function is used. Defaults to FALSE.

Tol

For method = "Mestimator" only. The tolerance parameter used in the numerical integration procedure. Defaults to 1e-05.

k1

For biascorr = TRUE only. The value of $k_1$ in the definition of the bias-corrected estimator of the stable tail dependence function.

tau

For biascorr = TRUE only. The parameter of the power kernel.

startingValue

Initial value of the parameters in the minimization routine. Defaults to $c(1,1.5)$ if isotropic = TRUE and $c(1, 1.5, 0.75, 0.75)$ if isotropic = FALSE.

Omega

A $q$ x $q$ matrix specifying the metric with which the distance between the parametric and nonparametric estimates will be computed. The default is the identity matrix, i.e., the Euclidean metric.

iterate

A Boolean variable. If TRUE, then for method = "Mestimator" the estimator is calculated twice, first with Omega specified by the user, and then a second time with the optimal Omega calculated at the initial estimate. If method = "WLS", then continuous updating is used.

covMat

A Boolean variable. If TRUE (the default), the covariance matrix is calculated. Standard errors are obtained by taking the square root of the diagonal elements.

Value

A list with the following components:

theta

The estimator using the optimal weight matrix.

theta_pilot

The estimator without the optimal weight matrix.

covMatrix

The estimated covariance matrix for the estimator.

value

The value of the minimized function at theta.

Details

The parameters of the Brown-Resnick process are either $(\alpha,\rho)$ for an isotropic process or $(\alpha,\rho,\beta,c)$ for an anisotropic process. The matrix indices can be either user-defined or returned from the function selectGrid with cst = c(0,1). Estimation might be rather slow when iterate = TRUE or even when covMat = TRUE.

References

Einmahl, J.H.J., Kiriliouk, A., and Segers, J. (2016). A continuous updating weighted least squares estimator of tail dependence in high dimensions. See http://arxiv.org/abs/1601.04826.

Einmahl, J.H.J., Kiriliouk, A., Krajina, A., and Segers, J. (2016). An Mestimator of spatial tail dependence. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(1), 275-298.

See Also

selectGrid

Examples

## define the locations of 9 stations
## locations <- cbind(rep(c(1:3), each = 3), rep(1:3, 3))
## select the pairs of locations
## indices <- selectGrid(cst = c(0,1), d = 9, locations = locations, maxDistance = 1.5)
## The Brown-Resnick process
## set.seed(1)
## x <- SpatialExtremes::rmaxstab(n = 1000, coord = locations, cov.mod = "brown",
##                               range = 3, smooth = 1)
## Calculate the estimtors.
## EstimationBR(x, locations, indices, 100, method = "Mestimator", isotropic = TRUE,
##             covMat = FALSE)$theta
## EstimationBR(x, locations, indices, 100, method = "WLS", isotropic = TRUE,
## covMat = FALSE)$theta