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 `\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`

.

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.

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]