ExpectMES {ExtremeRisks}R Documentation

Marginal Expected Shortfall Expectile Based Estimation


Computes a point and interval estimate of the Marginal Expected Shortfall (MES) using an expectile based approach.


ExpectMES(data, tau, tau1, method="LAWS", var=FALSE, varType="asym-Dep", bias=FALSE,
          bigBlock=NULL, smallBlock=NULL, k=NULL, alpha_n=NULL, alpha=0.05)



A vector of (1 \times n) observations.


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


A real in (0,1) specifying the extreme level \tau'_n. See Details.


A string specifying the method used to estimate the expecile. By default est="LAWS" specifies the use of the LAWS based estimator. See Details.


If var=TRUE then an estimate of the asymptotic variance of the MES estimator is computed.


A string specifying the type of asymptotic variance to compute. By default varType="asym-Dep" specifies the variance estimator for serial dependent observations. See Details.


A logical value. By default bias=FALSE specifies that no bias correction is computed. See Details.


An interger specifying the size of the big-block used to estimaste the asymptotic variance. See Details.


An interger specifying the size of the small-block used to estimaste the asymptotic variance. See Details.


An integer specifying the value of the intermediate sequence k_n. See Details.


A real in (0,1) specifying the quantile's extreme level to be use in order to estimate the expectile's extreme level.


A real in (0,1) specifying the confidence level (1-\alpha)100\% of the approximate confidence interval for the expecile at the intermedite level.


For a dataset data of sample size n, an estimate of the \tau'_n-th MES is computed. The estimation of the MES at the extreme level tau1 (\tau'_n) is indeed meant to be a prediction. Two estimators are available: the so-called Least Asymmetrically Weighted Squares (LAWS) based estimator and the Quantile-Based (QB) estimator. The definition of both estimators depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex for details). The observations can be either independent or temporal dependent. See Section 4 in Padoan and Stupfler (2020) for details.


A list with elements:


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.

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics to nancial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.

Drees, H. (2000). Weighted approximations of tail processes for \beta-mixing random variables. Annals of Applied Probability, 10, 1274-1301.

Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1989). Extremes and related properties of random sequences and processes. Springer.

See Also

QuantMES, HTailIndex, predExpectiles, extQuantile


# Marginl Expected Shortfall expectile based estimation at the extreme level
# obtained with 2-dimensional data simulated from an AR(1) with bivariate
# Student-t distributed innovations

tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"

# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=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 <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)

# Extreme MES expectile based estimation
MESHat <- ExpectMES(data, NULL, NULL, var=TRUE, k=150, bigBlock=bigBlock,
                    smallBlock=smallBlock, alpha_n=alpha_n)

[Package ExtremeRisks version 0.0.4 Index]