ExpectMES {ExtremeRisks}R Documentation

Marginal Expected Shortfall Expectile Based Estimation

Description

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

Usage

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)

Arguments

data

A vector of (1 x n) observations.

tau

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

tau1

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

method

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

var

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

varType

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.

bias

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

k

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

alpha_n

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

alpha

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

Details

For a dataset data of sample size n, an estimate of the τ'_n-th MES is computed. The estimation of the MES at the extreme level tau1 (τ'_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 γ. Here, γ 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.

Value

A list with elements:

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@ensai.fr, http://ensai.fr/en/equipe/stupfler-gilles/

References

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 β-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

Examples

# 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)
MESHat


[Package ExtremeRisks version 0.0.4 Index]