extMultiQuantile {ExtremeRisks}R Documentation

Multidimensional Value-at-Risk (VaR) or Extreme Quantile (EQ) Estimation

Description

Computes point estimates and (1-α)100\% confidence regions for d-dimensional VaR based on the Weissman estimator.

Usage

extMultiQuantile(data, tau, tau1, var=FALSE, varType="asym-Ind-Log", bias=FALSE,
                 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.

tau1

A real in (0,1) specifying the extreme level τ'_n. See Details.

var

If var=TRUE then an estimate of the asymptotic variance-covariance matrix of the d-dimensional VaR estimator is computed.

varType

A string specifying the type of asymptotic variance-covariance matrix to compute. By default varType="asym-Ind-Log" specifies that the variance-covariance matrix is obtained assuming dependent variables and exploiting the logarithmic scale. See Details.

bias

A logical value. By default biast=FALSE specifies that no bias correction is computed. 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 VaR.

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, the VaR or EQ, correspoding to the extreme level tau1, is computed by applying the d-dimensional Weissman estimator. The definition of the Weissman estimator depends on the estimation of the d-dimensional tail index γ. Here, γ is estimated using the Hill estimation (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

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics to financial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.

de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer-Verlag, New York.

See Also

MultiHTailIndex, estMultiExpectiles, predMultiExpectiles

Examples

# Extreme quantile estimation at the extreme level tau1 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 <- 0.95
# Extreme level (or tail probability 1-tau1 of unobserved quantile)
tau1 <- 0.9995

# 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
extQHat <- extMultiQuantile(data, tau, tau1, TRUE)

extQHat$ExtQHat
extQHat$VarCovExQHat
# run the following command to see the graphical representation

 extQHat <- extMultiQuantile(data, tau, tau1, TRUE, plot=TRUE)


[Package ExtremeRisks version 0.0.4 Index]