EstimationGumbel {tailDepFun}

Estimation of the parameter of the Gumbel model

Description

Estimation the parameter of the Gumbel model, using either the pairwise M-estimator or weighted least squares (WLS).

Usage

EstimationGumbel(
  x,
  indices,
  k,
  method,
  biascorr = FALSE,
  k1 = (nrow(x) - 10),
  tau = 5,
  covMat = TRUE
)

Arguments

x

An n x d data matrix.

indices

A q x d matrix containing at least 2 non-zero elements per row, representing the values in which we will evaluate the stable tail dependence function. For method = Mestimator, this matrix should contain exactly two ones per row.

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.

biascorr

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

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.

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.

Details

The matrix indices can be either user defined or returned by selectGrid. For method = "Mestimator", only a grid with exactly two ones per row is accepted, representing the pairs to be used.

Value

For WLS, a list with the following components:

theta	The estimator with weight matrix identity.
covMatrix	The estimated covariance matrix for the estimator.
value	The value of the minimized function at theta.

References

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.

See Also

selectGrid

Examples

## Generate data with theta = 0.5
set.seed(1)
n <- 1000
cop <- copula::gumbelCopula(param = 2, dim = 3)
data <- copula::rCopula(n = n,copula = cop)
## Transform data to unit Pareto margins
x <- apply(data, 2, function(i) n/(n + 0.5 - rank(i)))
## Define indices in which we evaluate the estimator
indices <- selectGrid(c(0,1), d = 3)
EstimationGumbel(x, indices, k = 50, method = "WLS", biascorr = TRUE)