estExtLevel {ExtremeRisks} | R Documentation |
Extreme Level Estimation
Description
Estimates the expectile's extreme level corresponding to a quantile's extreme level.
Usage
estExtLevel(alpha_n, data=NULL, gammaHat=NULL, VarGamHat=NULL, tailest="Hill", k=NULL,
var=FALSE, varType="asym-Dep", bigBlock=NULL, smallBlock=NULL, alpha=0.05)
Arguments
alpha_n |
A real in |
data |
A vector of |
gammaHat |
A real specifying an estimate of the tail index. By default |
VarGamHat |
A real specifying an estimate of the variance of the tail index estimate. By default |
tailest |
A string specifying the type of tail index estimator to be used. By default |
k |
An integer specifying the value of the intermediate sequence |
var |
If |
varType |
A string specifying the asymptotic variance to compute. By default |
bigBlock |
An interger specifying the size of the big-block used to estimaste the asymptotic variance. See Details. |
smallBlock |
An interger specifying the size of the small-block used to estimaste the asymptotic variance. See Details. |
alpha |
A real in |
Details
For a given extreme level for the
-th quantile, an estimate of the extreme level
is computed such that
. The estimator is defined by
where is a consistent estimator of the tail index
. If a value for the parameter
gammaHat
is given, then such a value is used to compute . If
gammaHat
is NULL
and a dataset is provided through the parameter data
, then the tail index is estimated by a suitable estimator
. See Section 6 in Padoan and Stupfler (2020) for more details.
If
VarGamHat
is specified, i.e. the variance of the tail index estimator, then the variance of the extreme level estimatoris computed by using such value.
When estimating the tail index, if
tailest='Hill'
thenis estimated using the Hill estimator (see also HTailIndex). If
tailest='ML'
thenis estimated using the Maximum Likelihood estimator (see MLTailIndex). If
tailest='ExpBased'
thenis estimated using the expectile based estimator (see EBTailIndex). If
tailest='Moment'
thenis estimated using the moment based estimator (see MomTailIndex). See Padoan and Stupfler (2020) for details.
-
k
oris the value of the so-called intermediate sequence
,
. Its represents a sequence of positive integers such that
and
as
. Practically, when
tailest="Hill"
then the valuespecifies the number of
k
larger order statistics to be used to estimate
by the Hill estimator. See MLTailIndex, EBTailIndex and MomTailIndex for the other estimators.
If
var=TRUE
then the asymptotic variance of the extreme level estimator is computed by applying the delta method, i.e.where
is provided by
VarGamHat
or is estimated when esitmating the tail index throughtailest='Hill'
andtailest='ML'
. See HTailIndex and MLTailIndex for details on how the variance is computed.Given a small value
then an asymptotic confidence interval for the extreme level,
, with approximate nominal confidence level
is computed.
Value
A list with elements:
-
tauHat
: an estimate of the extreme level;
-
tauVar
: an estimate of the asymptotic variance of the extreme level estimator;
-
tauCI
: an estimate of the approximateconfidence interval for the extreme level
.
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). Extreme expectile estimation for heavy-tailed time series. arXiv e-prints arXiv:2004.04078, https://arxiv.org/abs/2004.04078.
Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.
See Also
estExpectiles, predExpectiles, extQuantile
Examples
# Extreme level estimation for a given quantile's extreme level alpha_n
# obtained with 1-dimensional data simulated from an AR(1) with Student-t innovations
tsDist <- "studentT"
tsType <- "AR"
# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)
# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15
# quantile's extreme level
alpha_n <- 0.999
# sample size
ndata <- 2500
# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)
# expectile's extreme level estimation
tau1Hat <- estExtLevel(alpha_n, data, var=TRUE, k=150, bigBlock=bigBlock,
smallBlock=smallBlock)
tau1Hat