Function AsymVar

Description

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

Usage

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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.

Value

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.

See Also

Mestimator, selectPairIndices, pairCoordinates

Examples

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## 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)