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 (0,1) specifying the extreme level α_n for the quantile. See Details.

data

A vector of (1 x 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-α)100\% of the approximate confidence interval for the expecile at the intermedite level.

Details

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

\hat{τ}_n'(α_n) = 1 - (1 - α_n)\frac{\hat{γ}_n}{1-\hat{γ}_n}

where \hat{γ}_n is a consistent estimator of the tail index γ. If a value for the parameter gammaHat is given, then such a value is used to compute \hat{τ}_n'. If gammaHat is NULL and a dataset is provided through the parameter data, then the tail index γ is estimated by a suitable estimator \hat{γ}_n. See Section 6 in Padoan and Stupfler (2020) for more details.

Value

A list with elements:

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

[Package ExtremeRisks version 0.0.4 Index]