# Extreme Level Estimation Estimates the expectile's extreme level corresponding to a quantile's extreme level. ```r estExtLevel(alpha_n, data=NULL, gammaHat=NULL, VarGamHat=NULL, tailest="Hill", k=NULL, var=FALSE, varType="asym-Dep", bigBlock=NULL, smallBlock=NULL, alpha=0.05) ``` ## Arguments - `alpha_n`: A real in $(0,1)$ specifying the extreme level $\alpha_n$ for the quantile. See Details . - `data`: A vector of $(1 x n)$ observations to be used to estimate the tail index in the case it is not provided. By default `data=NULL` specifies that no data are given. - `gammaHat`: A real specifying an estimate of the tail index. By default `gammaHat=NULL` specifies that no estimate is given. See Details . - `VarGamHat`: A real specifying an estimate of the variance of the tail index estimate. By default `VarGamHat=NULL` specifies that no estimate is given. See Details . - `tailest`: A string specifying the type of tail index estimator to be used. By default `tailest="Hill"` specifies the use of Hill estimator. See Details . - `k`: An integer specifying the value of the intermediate sequence $k_n$. See Details . - `var`: If `var=TRUE` then an estimate of the variance of the extreme level estimator is computed. - `varType`: A string specifying the asymptotic variance to compute. By default `varType="asym-Dep"` specifies the variance estimator for serial dependent observations. See Details . - `bigBlock`: An interger specifying the size of the big-block used to estimaste the asymptotic variance. See Details . - `smallBlock`: An interger specifying the size of the small-block used to estimaste the asymptotic variance. See Details . - `alpha`: A real in $(0,1)$ specifying the confidence level $(1-\alpha)100\%$ of the approximate confidence interval for the expecile at the intermedite level. ## Details For a given extreme level $\alpha_n$ for the $\alpha_n$-**th** quantile, an estimate of the extreme level $\tau_n'(\alpha_n)$ is computed such that $\xi_{\tau_n'(\alpha_n)}=q_{\alpha_n}$. The estimator is defined by $\hat{\tau}_n'(\alpha_n) = 1 - (1 - \alpha_n)\frac{\hat{\gamma}_n}{1-\hat{\gamma}_n}$ where $\hat{\gamma}_n$ is a consistent estimator of the tail index $\gamma$. If a value for the parameter `gammaHat` is given, then such a value is used to compute $\hat{\tau}_n'$. If `gammaHat` is `NULL` and a dataset is provided through the parameter `data`, then the tail index $\gamma$ is estimated by a suitable estimator $\hat{\gamma}_n$. See Section 6 in Padoan and Stupfler (2020) for more details. * If `VarGamHat` is specified, i.e. the variance of the tail index estimator, then the variance of the extreme level estimator $\hat{\tau}_n'$ is computed by using such value. * When estimating the tail index, if `tailest='Hill'` then $\gamma$ is estimated using the Hill estimator (see also HTailIndex ). If `tailest='ML'` then $\gamma$ is estimated using the Maximum Likelihood estimator (see MLTailIndex ). If `tailest='ExpBased'` then $\gamma$ is estimated using the expectile based estimator (see EBTailIndex ). If `tailest='Moment'` then $\gamma$ is estimated using the moment based estimator (see MomTailIndex ). See Padoan and Stupfler (2020) for details. * `k` or $k_n$ is the value of the so-called intermediate sequence $k_n$, $n=1,2,...$. Its represents a sequence of positive integers such that $k_n -\> \infty$ and $k_n/n -\> 0$ as $n -\> \infty$. Practically, when `tailest="Hill"` then the value $k_n$ specifies the number of `k`$+1$ larger order statistics to be used to estimate $\gamma$ by the Hill estimator. See MLTailIndex , EBTailIndex and MomTailIndex for the other estimators. * If `var=TRUE` then the asymptotic variance of the extreme level estimator is computed by applying the delta method, i.e. $Var(\tau_n') = Var(\hat{\gamma}_n) * (\alpha_n-1)^2 / (1-\hat{\gamma}_n)^4$ where $Var(\hat{\gamma}_n$ is provided by `VarGamHat` or is estimated when esitmating the tail index through `tailest='Hill'` and `tailest='ML'`. See HTailIndex and MLTailIndex for details on how the variance is computed. * Given a small value $\alpha\in (0,1)$ then an asymptotic confidence interval for the extreme level, $\tau_n'(\alpha_n)$, with approximate nominal confidence level $(1-\alpha)100\%$ is computed. ## Returns A list with elements: * `tauHat`: an estimate of the extreme level $\tau_n'$; * `tauVar`: an estimate of the asymptotic variance of the extreme level estimator $\hat{\tau}_n'(\alpha_n)$; * `tauCI`: an estimate of the approximate $(1-\alpha)100\%$ confidence interval for the extreme level $\tau_n'(\alpha_n)$. ## References Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, **Journal of Business & Economic Statistics**, 41 (3) 876-889. 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. ## Author(s) Simone Padoan, simone.padoan@unibocconi.it , [https://faculty.unibocconi.it/simonepadoan/](https://faculty.unibocconi.it/simonepadoan/); Gilles Stupfler, gilles.stupfler@univ-angers.fr , [https://math.univ-angers.fr/~stupfler/](https://math.univ-angers.fr/~stupfler/) ## See Also estExpectiles , predExpectiles , extQuantile ## Examples ```r # Extreme level estimation for a given quantile's extreme level alpha_n # obtained with 1-dimensional data simulated from an AR(1) with Student-t innovations tsDist <- "studentT" tsType <- "AR" # parameter setting corr <- 0.8 df <- 3 par <- c(corr, df) # Big- small-blocks setting bigBlock <- 65 smallBlock <- 15 # quantile's extreme level alpha_n <- 0.999 # sample size ndata <- 2500 # Simulates a sample from an AR(1) model with Student-t innovations data <- rtimeseries(ndata, tsDist, tsType, par) # expectile's extreme level estimation tau1Hat <- estExtLevel(alpha_n, data, var=TRUE, k=150, bigBlock=bigBlock, smallBlock=smallBlock) tau1Hat ```

estExtLevel function