estExtLevel {ExtremeRisks}

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 (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.

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.

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).

Author(s)

Simone Padoan, [e-mail], http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, [e-mail], 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