extMultiQuantile {ExtremeRisks} | R Documentation |
Multidimensional Value-at-Risk (VaR) or Extreme Quantile (EQ) Estimation
Description
Computes point estimates and 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 |
tau |
A real in |
tau1 |
A real in |
var |
If |
varType |
A string specifying the type of asymptotic variance-covariance matrix to compute. By default |
bias |
A logical value. By default |
k |
An integer specifying the value of the intermediate sequence |
alpha |
A real in |
plot |
A logical value. By default |
Details
For a dataset data
of d
-dimensional observations and sample size , 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 . Here,
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.
The so-called intermediate level
tau
oris a sequence of positive reals such that
as
. Practically, for each variable,
is a small proportion of observations in the observed data sample that exceed the
-th empirical quantile. Such proportion of observations is used to estimate the individual
-th quantile and tail index
.
The so-called extreme level
tau1
oris a sequence of positive reals such that
as
. For each variable, the value
is meant to be a small tail probability such that
or
. It is also assumed that
as
, where
is a positive finite constant. The value
is the expected number of exceedances of the individual
-th quantile. Typically,
which means that it is expected that there are no observations in a data sample exceeding the individual quantile of level
.
If
var=TRUE
then an estimate of the asymptotic variance-covariance matrix of the-th
d
-dimensional quantile is computed. The data are regarded as temporal independent observations coming from dependent variables. The asymptotic variance-covariance matrix is estimated exploiting the formula in Section 5 of Padoan and Stupfler (2020). In particular, the variance-covariance matrix is computed exploiting the asymptotic behaviour of the normalized quantile estimator which is expressed in logarithmic scale. This is achieved throughvarType="asym-Ind-Log"
. IfvarType="asym-Ind"
then the variance-covariance matrix is computed exploiting the asymptotic behaviour of thed
-dimensional relative quantile estimator appropriately normalized (see formula in Section 5 of Padoan and Stupfler (2020)).If
bias=TRUE
then an estimate of each individual–th quantile is estimated using the formula in page 330 of de Haan et al. (2016), which provides a bias corrected version of the Weissman estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. For simplicity standard the variance-covariance matrix is still computed using formula in Section 5 of Padoan and Stupfler (2020).
-
k
oris the value of the so-called intermediate sequence
,
. Its represents a sequence of positive integers such that
and
as
. Practically, for each marginal distribution, the value
specifies the number of
k
larger order statistics to be used to estimate the individual
-th empirical quantile and individual tail index
for
. The intermediate level
can be seen defined as
.
Given a small value
then an estimate of an asymptotic confidence region for
-th
d
-dimensional quantile, with approximate nominal confidence level, is computed. The confidence regions are computed exploiting the asymptotic behaviour of the normalized quantile estimator in logarithmic scale. This is an "asymmetric" region and it is achieved through
varType="asym-Ind-Log"
. A "symmetric" region is obtained exploiting the asymptotic behaviour of the relative quantile estimator appropriately normalized, see formula in Section 5 of Padoan and Stupfler (2020). This is achieved throughvarType="asym-Ind"
.If
plot=TRUE
then a graphical representation of the estimates is not provided.
Value
A list with elements:
-
ExtQHat
: an estimate of thed
-dimensional VaR or-th
d
-dimensional quantile; -
VarCovExQHat
: an estimate of the asymptotic variance-covariance of thed
-dimensional VaR estimator; -
EstConReg
: an estimate of the approximateconfidence regions for the
d
-dimensional VaR.
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)