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 (1 x n) observations. |
tau |
A real in (0,1) specifying the intermediate level τ_n. See Details\. |
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