estExtLevel {ExtremeRisks}R Documentation

Extreme Level Estimation


Estimates the expectile's extreme level corresponding to a quantile's extreme level.


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)



A real in (0,1) specifying the extreme level \alpha_n for the quantile. See Details.


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.


A real specifying an estimate of the tail index. By default gammaHat=NULL specifies that no estimate is given. See Details.


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.


A string specifying the type of tail index estimator to be used. By default tailest="Hill" specifies the use of Hill estimator. See Details.


An integer specifying the value of the intermediate sequence k_n. See Details.


If var=TRUE then an estimate of the variance of the extreme level estimator is computed.


A string specifying the asymptotic variance to compute. By default varType="asym-Dep" specifies the variance estimator for serial dependent observations. See Details.


An interger specifying the size of the big-block used to estimaste the asymptotic variance. See Details.


An interger specifying the size of the small-block used to estimaste the asymptotic variance. See Details.


A real in (0,1) specifying the confidence level (1-\alpha)100\% of the approximate confidence interval for the expecile at the intermedite level.


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.


A list with elements:


Simone Padoan,,; Gilles Stupfler,,


Padoan A.S. and Stupfler, G. (2020). Extreme expectile estimation for heavy-tailed time series. arXiv e-prints arXiv: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


# 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,

[Package ExtremeRisks version 0.0.4 Index]