EstimationBR: Estimation of the parameters of the Brown-Resnick process
In tailDepFun: Minimum Distance Estimation of Tail Dependence Models

Description Usage Arguments Details Value References See Also Examples

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

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
)

`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.

The parameters of the Brown-Resnick process are either (α,ρ) for an isotropic process or (α,ρ,β,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.

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`.

Einmahl, J.H.J., Kiriliouk, A., and Segers, J. (2018). A continuous updating weighted least squares estimator of tail dependence in high dimensions. Extremes 21(2), 205-233.

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.

selectGrid

## 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