EstimationMaxLinear: Estimation of the parameters of the max-linear model
In tailDepFun: Minimum Distance Estimation of Tail Dependence Models

Description Usage Arguments Details Value References See Also Examples

Estimation the parameters of the max-linear model, using either the pairwise M-estimator or weighted least squares (WLS).

EstimationMaxLinear(
  x,
  indices,
  k,
  method,
  Bmatrix = NULL,
  Ldot = NULL,
  biascorr = FALSE,
  k1 = (nrow(x) - 10),
  tau = 5,
  startingValue,
  Omega = diag(nrow(indices)),
  iterate = FALSE,
  covMat = TRUE,
  GoFtest = FALSE,
  dist = 0.01,
  EURO = FALSE
)

`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.
`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`.
`Bmatrix`	A function that converts the parameter vector theta to a parameter matrix B. If nothing is provided, then a simple 2-factor model is assumed.
`Ldot`	For `method = "WLS"` only. A q x p matrix, where p is the number of parameters of the model. Represents the total derivative of the function L defined in Einmahl et al. (2016). If nothing is provided, then a simple 2-factor model is assumed.
`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.
`startingValue`	Initial value of the parameters in the minimization routine.
`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. For `method = "WLS"` only. If `TRUE`, then continuous updating is used. Defaults to `FALSE`.
`covMat`	A Boolean variable. For `method = "WLS"` only. If `TRUE` (the default), the covariance matrix is calculated.
`GoFtest`	A Boolean variable. For `method = "WLS"` only. If `TRUE`, then the goodness-of-fit test of Corollary 2.6 from Einmahl et al. (2016) is performed. Defaults to `FALSE`.
`dist`	A positive scalar. If `GoFtest = TRUE`, only eigenvalues ν larger than `dist` are used; see Corollary 2.6 (Einmahl et al., 2016). Defaults to 0.01.
`EURO`	A Boolean variable. If `TRUE`, then the model from Einmahl et al. (2016, Section 4) is assumed, and the corresponding `Bmatrix` and `Ldot` are used.

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. The functions Bmatrix and Ldot can be defined such that they represent a max-linear model on a directed acyclic graph: see the vignette for this package for an example.

For Mestimator, the estimator theta is returned. For WLS, a list with the following components:

`theta`	The estimator with estimated 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`.
`GoFresult`	A list of length two, returning the value of the test statistic and `s`.

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.

selectGrid

## Generate data
set.seed(1)
n <- 1000
fr <- matrix(-1/log(runif(2*n)), nrow = n, ncol = 2)
data <- cbind(pmax(0.3*fr[,1],0.7*fr[,2]),pmax(0.5*fr[,1],0.5*fr[,2]),pmax(0.9*fr[,1],0.1*fr[,2]))
## 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(cst = c(0,0.5,1), d = 3)
EstimationMaxLinear(x, indices, k = 100, method = "WLS", startingValue = c(0.3,0.5,0.9))
indices <- selectGrid(cst = c(0,1), d = 3)
EstimationMaxLinear(x, indices, k = 100, method = "Mestimator", startingValue = c(0.3,0.5,0.9))