ExtremeRisks (version 0.0.4)

EBTailIndex: Expectile Based Tail Index Estimation

Description

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

Usage

EBTailIndex(data, tau, est=NULL)

Value

An estimate of the tain index \(\gamma\).

Arguments

data

A vector of \((1 \times n)\) observations.

tau

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

est

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

Author

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

Details

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.

• The so-called intermediate level tau or \(\tau_n\) is a sequence of positive reals such that \(\tau_n \to 1\) as \(n \to \infty\). Practically, \(\tau_n \in (0,1)\) is the ratio between the empirical mean distance of the \(\tau_n\)-th expectile from the smaller observations and the empirical mean distance of of the \(\tau_n\)-th expectile from all the observations. An estimate of \(\tau_n\)-th expectile is computed and used in turn to estimate \(\gamma\).

• The value est, if provided, is meant to be an esitmate of the \(\tau_n\)-th expectile which is used to estimate \(\gamma\). On the contrary, if est=NULL, then the routine EBTailIndex estimate first the \(\tau_n\)-th expectile expectile and then use it to estimate \(\gamma\).

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

HTailIndex, MomTailIndex, MLTailIndex,

Examples

# 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)
gammaHat