estExtLevel: 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)

Value

A list with elements:

tauHat: an estimate of the extreme level \(\tau_n'\);
tauVar: an estimate of the asymptotic variance of the extreme level estimator \(\hat{\tau}_n'(\alpha_n)\);
tauCI: an estimate of the approximate \((1-\alpha)100\%\) confidence interval for the extreme level \(\tau_n'(\alpha_n)\).

Arguments

alpha_n: A real in \((0,1)\) specifying the extreme level \(\alpha_n\) for the quantile. See Details.
data: A vector of \((1 \times n)\) observations to be used to estimate the tail index in the case it is not provided. By default data=NULL specifies that no data are given.
gammaHat: A real specifying an estimate of the tail index. By default gammaHat=NULL specifies that no estimate is given. See Details.
VarGamHat: A real specifying an estimate of the variance of the tail index estimate. By default VarGamHat=NULL specifies that no estimate is given. See Details.
tailest: A string specifying the type of tail index estimator to be used. By default tailest="Hill" specifies the use of Hill estimator. See Details.
k: An integer specifying the value of the intermediate sequence \(k_n\). See Details.
var: If var=TRUE then an estimate of the variance of the extreme level estimator is computed.
varType: A string specifying the asymptotic variance to compute. By default varType="asym-Dep" specifies the variance estimator for serial dependent observations. See Details.
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 \((0,1)\) specifying the confidence level \((1-\alpha)100\%\) of the approximate confidence interval for the expecile at the intermedite level.

Author

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

Details

For a given extreme level \(\alpha_n\) for the \(\alpha_n\)-th quantile, an estimate of the extreme level \(\tau_n'(\alpha_n)\) is computed such that \(\xi_{\tau_n'(\alpha_n)}=q_{\alpha_n}\). The estimator is defined by

\(\hat{\tau}_n'(\alpha_n) = 1 - (1 - \alpha_n)\frac{\hat{\gamma}_n}{1-\hat{\gamma}_n}\)

where \(\hat{\gamma}_n\) is a consistent estimator of the tail index \(\gamma\). If a value for the parameter gammaHat is given, then such a value is used to compute \(\hat{\tau}_n'\). If gammaHat is NULL and a dataset is provided through the parameter data, then the tail index \(\gamma\) is estimated by a suitable estimator \(\hat{\gamma}_n\). 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 estimator \(\hat{\tau}_n'\) is computed by using such value.
When estimating the tail index, if tailest='Hill' then \(\gamma\) is estimated using the Hill estimator (see also HTailIndex). If tailest='ML' then \(\gamma\) is estimated using the Maximum Likelihood estimator (see MLTailIndex). If tailest='ExpBased' then \(\gamma\) is estimated using the expectile based estimator (see EBTailIndex). If tailest='Moment' then \(\gamma\) is estimated using the moment based estimator (see MomTailIndex). See Padoan and Stupfler (2020) for details.
k or \(k_n\) is the value of the so-called intermediate sequence \(k_n\), \(n=1,2,\ldots\). Its represents a sequence of positive integers such that \(k_n \to \infty\) and \(k_n/n \to 0\) as \(n \to \infty\). Practically, when tailest="Hill" then the value \(k_n\) specifies the number of k\(+1\) larger order statistics to be used to estimate \(\gamma\) 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.

\(Var(\tau_n') = Var(\hat{\gamma}_n) * (\alpha_n-1)^2 / (1-\hat{\gamma}_n)^4\)

where \(Var(\hat{\gamma}_n\) is provided by VarGamHat or is estimated when esitmating the tail index through tailest='Hill' and tailest='ML'. See HTailIndex and MLTailIndex for details on how the variance is computed.
Given a small value \(\alpha\in (0,1)\) then an asymptotic confidence interval for the extreme level, \(\tau_n'(\alpha_n)\), with approximate nominal confidence level \((1-\alpha)100\%\) is computed.

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.

Examples

Run this code

# 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

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