EBTailIndex {ExtremeRisks} | R Documentation |

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

EBTailIndex(data, tau, est=NULL)

`data` |
A vector of |

`tau` |
A real in |

`est` |
A real specifying the estimate of the expectile at the intermediate level |

For a dataset `data`

of sample size *n*, the tail index *γ* of its (marginal) distribution is estimated using the EB estimator:

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

where *\hat{\bar{F}}_n* is the empirical survival function of the observations, *tilde{xi}_{tau_n}* is an estimate of the *τ_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*τ_n -> 1*as*n -> ∞*. Practically,*τ_n in (0,1)*is the ratio between the empirical mean distance of the*τ_n*-*th*expectile from the smaller observations and the empirical mean distance of of the*τ_n*-*th*expectile from all the observations. An estimate of*τ_n*-*th*expectile is computed and used in turn to estimate*γ*.The value

`est`

, if provided, is meant to be an esitmate of the*τ_n*-*th*expectile which is used to estimate*γ*. On the contrary, if`est=NULL`

, then the routine`EBTailIndex`

estimate first the*τ_n*-*th*expectile expectile and then use it to estimate*γ*.

An estimate of the tain index *γ*.

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.

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) gammaHat

[Package *ExtremeRisks* version 0.0.4 Index]