Asymptotic variance matrix for the Brown-Resnick process.

Description

Computes the asymptotic variance matrix for the Brown-Resnick process, estimated using the pairwise M-estimator or the weighted least squares estimator.

Usage

AsymVarBR(locations, indices, par, method, Tol = 1e-05)

Arguments

Details

The parameters of a The matrix indices can be either user-defined or returned from the function selectGrid with cst = c(0,1). Calculation might be rather slow for method = "Mestimator".

Value

A q by q matrix.

References

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.

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.

See Also

selectGrid

Examples

locations <- cbind(rep(1:2, 3), rep(1:3, each = 2))
indices <- selectGrid(cst = c(0,1), d = 6, locations = locations, maxDistance = 1)
AsymVarBR(locations, indices, par = c(1.5,3), method = "WLS")

Asymptotic variance matrix for the Gumbel model.

Description

Computes the asymptotic variance matrix for the Gumbel model, estimated using the pairwise M-estimator or the weighted least squares estimator.

Usage

AsymVarGumbel(indices, par, method)

Arguments

Details

The matrix indices can be either user defines 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

A q by q matrix.

References

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.

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.

See Also

selectGrid

Examples

indices <- selectGrid(c(0,1), d = 3, nonzero = c(2,3))
AsymVarGumbel(indices, par = 0.7, method = "WLS")

Asymptotic variance matrix for the max-linear model.

Description

Computes the asymptotic variance matrix for the max-linear model, estimated using the weighted least squares estimator.

Usage

AsymVarMaxLinear(indices, par, Bmatrix = NULL)

Arguments

Value

A q by q matrix.

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.

See Also

selectGrid

Examples

indices <- selectGrid(c(0,0.5,1), d = 3, nonzero = 3)
AsymVarMaxLinear(indices, par = c(0.1,0.55,0.75))

Estimation of the parameters of the Brown-Resnick process

Description

Estimation the parameters of the Brown-Resnick process, using either the pairwise M-estimator or weighted least squares (WLS).

Usage

EstimationBR(
  x,
  locations,
  indices,
  k,
  method,
  isotropic = FALSE,
  biascorr = FALSE,
  Tol = 1e-05,
  k1 = (nrow(x) - 10),
  tau = 5,
  startingValue = NULL,
  Omega = diag(nrow(indices)),
  iterate = FALSE,
  covMat = TRUE
)

Arguments

Details

The parameters of the Brown-Resnick process are either (\alpha,\rho) for an isotropic process or (\alpha,\rho,\beta,c) for an anisotropic process. The matrix indices can be either user-defined or returned from the function selectGrid with cst = c(0,1). Estimation might be rather slow when iterate = TRUE or even when covMat = TRUE.

Value

A list with the following components:

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

## define the locations of 9 stations
locations <- cbind(rep(c(1:3), each = 3), rep(1:3, 3))
## select the pairs of locations
indices <- selectGrid(cst = c(0,1), d = 9, locations = locations, maxDistance = 1.5)
## The Brown-Resnick process
set.seed(1)
x <- SpatialExtremes::rmaxstab(n = 1000, coord = locations, cov.mod = "brown",
                               range = 3, smooth = 1)
## Calculate the estimtors.
EstimationBR(x, locations, indices, 100, method = "Mestimator", isotropic = TRUE,
             covMat = FALSE)$theta
EstimationBR(x, locations, indices, 100, method = "WLS", isotropic = TRUE,
covMat = FALSE)$theta

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

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:

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)

Estimation of the parameters of the max-linear model

Description

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

Usage

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
)

Arguments

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. 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.

Value

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

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.

See Also

selectGrid

Examples

## 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))

EUROSTOXX50 weekly negative log-returns.

Description

The first three columns represent the weekly negative log-returns of the index prices of the EUROSTOXX50 and of its subindices correspoding to the supersectors chemicals and insurance. The fourth and fifth columns represent the weekly negative log-returns of the index prices of the DAX and the CAC40 indices. The sixth to tenth columnds represent the weekly negative log-returns of the stock prices of Bayer, BASF, Allianz, AXA, and Airliquide respectively.

Format

dataEUROSTOXX is a matrix with 711 rows and 10 columns.

Source

Yahoo Finance

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.

Examples

data(dataEUROSTOXX)
## Transform data to unit Pareto margins
n <- nrow(dataEUROSTOXX)
x <- apply(dataEUROSTOXX, 2, function(i) n/(n + 0.5 - rank(i)))
## Define indices in which we evaluate the estimator
indices <- selectGrid(c(0,0.5,1), d = 10, nonzero = c(2,3))
start <- c(0.67,0.8,0.77,0.91,0.41,0.47,0.25,0.7,0.72,0.19,0.37,0.7,0.09,0.58)
## Estimate the parameters. Lasts up to ten minutes.

EstimationMaxLinear(x, indices, k = 40, method = "WLS", startingValue = start,
covMat = FALSE, EURO = TRUE)

Wind speeds in the Netherlands.

Description

Daily maximal speeds of wind gusts, measured in 0.1 m/s. The data are observed at 22 inland weather stations in the Netherlands. Only the summer months are presented here (June, July, August). Also included are the Euclidian coordinates of the 22 weather stations, where a distance of 1 corresponds to 100 kilometers.

Format

dataKNMI$data is a matrix with 672 rows and 22 columns, dataKNMI$loc is a matrix with 22 rows and 2 columns.

Source

KNMI

References

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.

Examples

data(dataKNMI)
n <- nrow(dataKNMI$data)
locations <- dataKNMI$loc
x <- apply(dataKNMI$data, 2, function(i) n/(n + 0.5 - rank(i)))
indices <- selectGrid(cst = c(0,1), d = 22, locations = locations, maxDistance = 0.5)
EstimationBR(x, locations, indices, k = 60, method = "Mestimator", isotropic = TRUE,
covMat = FALSE)$theta

Selects a grid of indices.

Description

Returns a regular grid of indices in which to evaluate the stable tail dependence function.

Usage

selectGrid(cst, d, nonzero = 2, locations = NULL, maxDistance = 10^6)

Arguments

Value

A matrix with q rows and d columns, where every row represents a vector in which we will evaluate the stable tail dependence function (for the weighted least squares estimator) or where every row indicates which pairs of variables to use (for the M-estimator)

Examples

selectGrid(cst = c(0,0.5,1), d = 3, nonzero = c(2,3))
locations <- cbind(rep(1:3, each = 3), rep(1:3,3))
selectGrid(c(0,1), d = 9, locations = locations, maxDistance = 1.5)

Empirical stable tail dependence function

Description

Returns the stable tail dependence function in dimension d, evaluated in a point cst.

Usage

stdfEmp(ranks, k, cst = rep(1, ncol(ranks)))

Arguments

Value

A scalar between \max(x_1,\ldots,x_d) and x_1 + \cdots + x_d.

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.

See Also

stdfEmpCorr

Examples

## Simulate data from the Gumbel copula and compute the extremal coefficient in dimension four.
set.seed(2)
cop <- copula::gumbelCopula(param = 2, dim = 4)
data <- copula::rCopula(n = 1000, copula = cop)
stdfEmp(apply(data,2,rank), k = 50)

Bias-corrected empirical stable tail dependence function

Description

Returns the bias-corrected stable tail dependence function in dimension d, evaluated in a point cst.

Usage

stdfEmpCorr(
  ranks,
  k,
  cst = rep(1, ncol(ranks)),
  tau = 5,
  k1 = (nrow(ranks) - 10)
)

Arguments

Details

The values for k1 and tau are chosen as recommended in Beirlant et al. (2016). This function might be slow for large n.

Value

A scalar between \max(x_1,\ldots,x_d) and x_1 + \cdots + x_d.

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.

Beirlant, J., Escobar-Bach, M., Goegebeur, Y., and Guillou, A. (2016). Bias-corrected estimation of stable tail dependence function. Journal of Multivariate Analysis, 143, 453-466.

See Also

stdfEmp

Examples

## Simulate data from the Gumbel copula
set.seed(2)
cop <- copula::gumbelCopula(param = 2, dim = 4)
data <- copula::rCopula(n = 1000, copula = cop)
stdfEmpCorr(apply(data,2,rank), k = 50)

Integrated empirical stable tail dependence function

Description

Analytical implementation of the integral of the bivariate stable tail dependence function over the unit square.

Usage

stdfEmpInt(ranks, k)

Arguments

Value

A positive scalar.

References

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.

Examples

ranks <- cbind(sample(1:20), sample(1:20))
stdfEmpInt(ranks, k = 5)

tailDepFun

Description

The package tailDepFun provides functions implementing two rank-based minimal distance estimation methods for parametric tail dependence models for distributions attracted to a max-stable law. The estimators, referred to as the pairwise M-estimator and the weighted least squares estimator, are described in Einmahl et al. (2016a) and Einmahl et al. (2016b). Extensive examples to illustrate the use of the package can be found in the accompanying vignette.

Details

Currently, this package allows for estimation of the Brown-Resnick process, the Gumbel (or logistic) model and max-linear models (possibly on a directed acyclic graph). The main functions of this package are EstimationBR, EstimationGumbel and EstimationMaxLinear, but several other functions are exported as well: stdfEmpInt returns the integral of the bivariate empirical stable tail dependence function over the unit square, and stdfEmp and stdfEmpCorr return the (bias-corrected) empirical stable tail dependence function. The functions AsymVarBR, AsymVarGumbel, AsymVarMaxLinear return the asymptotic covariance matrices of the estimators. An auxiliary function to select a regular grid of indices in which to evaluate the stable tail dependence function is exported as well, selectGrid. Finally, two datasets are available: dataKNMI (Einmahl et al., 2016) and dataEUROSTOXX (Einmahl et al., 2018).

References

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.

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.

Examples

## get a list of all help files of user-visible functions in the package
help(package = tailDepFun)