EBTailIndex {ExtremeRisks}R Documentation

Expectile Based Tail Index Estimation


Computes a point estimate of the tail index based on the Expectile Based (EB) estimator.


EBTailIndex(data, tau, est=NULL)



A vector of (1 \times n) observations.


A real in (0,1) specifying the intermediate level \tau_n. See Details\.


A real specifying the estimate of the expectile at the intermediate level tau.


For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is estimated using the EB estimator:

\hat{\gamma}_n^E =\left(1+\frac{\hat{\bar{F}}_n(\tilde{\xi}_{\tau_n})}{1-\tau_n}\right)^{-1} ,

where \hat{\bar{F}}_n is the empirical survival function of the observations, \tilde{\xi}_{\tau_n} is an estimate of the \tau_n-th expectile. The observations can be either independent or temporal dependent. See Padoan and Stupfler (2020) and Daouia et al. (2018) for details.


An estimate of the tain index \gamma.


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


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

HTailIndex, MomTailIndex, MLTailIndex,


# Tail index estimation based on the Expectile based estimator obtained with data
# simulated from an AR(1) with 1-dimensional Student-t distributed innovations

tsDist <- "studentT"
tsType <- "AR"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# Big- small-blocks setting
bigBlock <- 65
smallblock <- 15

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.97

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# tail index estimation
gammaHat <- EBTailIndex(data, tau)

[Package ExtremeRisks version 0.0.4 Index]