# AsymVar: Function 'AsymVar' In spatialTailDep: Estimation of spatial tail dependence models

## Description

Function to compute the asymptotic variance matrix of the pairwise M-estimator for the Smith model or the Brown-Resnick process.

## Usage

 `1` ```AsymVar(pairs, model, theta, Tol = 1e-05) ```

## Arguments

 `pairs` A q x 4 matrix giving the Cartesian coordinates of the q pairs of locations. `model` Choose between "smith" and "BR". `theta` Parameter vector. For the Smith model, `theta` must be equal to the 2 x 2 covariance matrix. For the Brown-Resnick pocess, `theta` = (α, ρ, β, c). `Tol` The tolerance in the numerical integration procedure. Defaults to 1e-5.

## Details

For a matrix of coordinates of pairs of locations, this function returns the asymptotic variance matrix of the estimator. An optimal weight matrix can be defined as the inverse of the asymptotic variance matrix. For a detailed description of this procedure, see Einmahl et al. (2014).

The parameter vector `theta` must be a positive semi-definite matrix if `model = "smith"` and a vector of length four if `model = "BR"`, where 0 < α < 1, ρ > 0, 0 < β ≤ π/2 and c > 0.

A q x q matrix.

## References

Einmahl, J.H.J., Kiriliouk, A., Krajina, A. and Segers, J. (2014), "An M-estimator of spatial tail dependence". See http://arxiv.org/abs/1403.1975.

`Mestimator`, `selectPairIndices`, `pairCoordinates`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```## Define the locations of three stations (locations <- rbind(c(1.0,1.0),c(2.0,1.0),c(1.2,2.5))) ## select pairs (pairIndices <- selectPairIndices(locations, maxDistance = 3)) (pairs <- pairCoordinates(locations, pairIndices)) ## Smith model parameter matrix (theta <- rbind(c(1.5, .5), c(.5, 1))) ## The matrix. Takes a couple of seconds to compute. ## AsymVar(pairs, model = "smith", theta = theta, Tol = 1e-04) ## Parameters of the Brown-Resnick process (theta <- c(1.5,1,0.5,0.25)) ## The matrix. Takes a couple of seconds to compute. ## AsymVar(pairs, model = "BR", theta = theta, Tol = 1e-04) ```