estMultiExpectiles {ExtremeRisks}R Documentation

Multidimensional High Expectile Estimation

Description

Computes point estimates and (1-α)100\% confidence regions for d-dimensional expectiles at the intermediate level.

Usage

estMultiExpectiles(data, tau, method="LAWS", tailest="Hill", var=FALSE,
                   varType="asym-Ind-Adj", k=NULL, alpha=0.05, plot=FALSE)

Arguments

data

A matrix of (n x d) observations.

tau

A real in (0,1) specifying the intermediate level τ_n. See Details.

method

A string specifying the method used to estimate the expecile. By default est="LAWS" specifies the use of the direct LAWS estimator. See Details.

tailest

A string specifying the type of tail index estimator. By default tailest="Hill" specifies the use of Hill estimator. See Details.

var

If var=TRUE then an estimate of the variance of the expectile estimator is computed.

varType

A string specifying the asymptotic variance-covariance matrix to compute. By default varType="asym-Ind-Adj" specifies that the variance-covariance matrix is computed assuming dependent variables and exploiting a suitable adjustment. See Details.

k

An integer specifying the value of the intermediate sequence k_n. See Details.

alpha

A real in (0,1) specifying the confidence level (1-α)100\% of the approximate confidence region for the d-dimensional expecile at the intermedite level.

plot

A logical value. By default plot=FALSE specifies that no graphical representation of the estimates is not provided. See Details.

Details

For a dataset data of d-dimensional observations and sample size n, an estimate of the τ_n-th d-dimensional is computed. Two estimators are available: the so-called direct Least Asymmetrically Weighted Squares (LAWS) and indirect Quantile-Based (QB). The QB estimator depends on the estimation of the d-dimensional tail index γ. Here, γ is estimated using the Hill estimator (see MultiHTailIndex). The data are regarded as d-dimensional temporal independent observations coming from dependent variables. See Padoan and Stupfler (2020) for details.

Value

A list with elements:

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@ensai.fr, http://ensai.fr/en/equipe/stupfler-gilles/

References

Padoan A.S. and Stupfler, G. (2020). Joint inference on extreme expectiles for multivariate heavy-tailed distributions. arXiv e-prints arXiv:2007.08944, https://arxiv.org/abs/2007.08944

See Also

MultiHTailIndex, predMultiExpectiles, extMultiQuantile

Examples

# Extreme expectile estimation at the intermediate level tau obtained with
# d-dimensional observations simulated from a joint distribution with
# a Gumbel copula and equal Frechet marginal distributions.
library(plot3D)
library(copula)
library(evd)

# distributional setting
copula <- "Gumbel"
dist <- "Frechet"

# parameter setting
dep <- 3
dim <- 3
scale <- rep(1, dim)
shape <- rep(3, dim)
par <- list(dep=dep, scale=scale, shape=shape, dim=dim)

# Intermediate level (or sample tail probability 1-tau)
tau <- .95

# sample size
ndata <- 1000

# Simulates a sample from a multivariate distribution with equal Frechet
# marginals distributions and a Gumbel copula
data <- rmdata(ndata, dist, copula, par)
scatter3D(data[,1], data[,2], data[,3])

# High d-dimensional expectile (intermediate level) estimation
expectHat <- estMultiExpectiles(data, tau, var=TRUE)
expectHat$ExpctHat
expectHat$VarCovEHat
# run the following command to see the graphical representation

 expectHat <- estMultiExpectiles(data, tau, var=TRUE, plot=TRUE)


[Package ExtremeRisks version 0.0.4 Index]