EstimationGumbel() R function from [tailDepFun]

Estimation of the parameter of the Gumbel model

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


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.

Returns

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` .

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.

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)

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.

EstimationGumbel function