QuantMES {ExtremeRisks}

Marginal Expected Shortfall Quantile Based Estimation

Description

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

Usage

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

Arguments

data

A vector of (1 \times n) observations.

tau

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

tau1

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

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 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. Estimates are obtained through the quantile based estimator defined in page 12 of Padoan and Stupfler (2020). Such an estimator 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.

• 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. See predExpectiles for details.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. See predExpectiles for details.

• If var=TRUE then an esitmate of the asymptotic variance of the tau'_n-th MES is computed. Notice that the estimation of the asymptotic variance is only available when \gamma is estimated using the Hill estimator (see HTailIndex). With independent observations the asymptotic variance is estimated by \hat{\gamma}^2, see Corollary 4.3 in Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Corollary 4.2 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". See Section 4 and 5 in Padoan and Stupfler (2020) for details. In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• If bias=TRUE then \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance is estimated by the formula in Corollary 3.8, with serial dependent observations, and \hat{\gamma}^2 with independent observation (see e.g. de Drees 2000, for the details).

• k or k_n is the value of the so-called intermediate sequence k_n, n=1,2,\ldots. Its represents a sequence of positive integers such that k_n \to \infty and k_n/n \to 0 as n \to \infty. k_n specifies the number of k+1 larger order statistics used in the definition of the Hill estimator (see HTailIndex for details).

• If the quantile's extreme level is provided by alpha_n, then expectile's extreme level tau'_n is replaced by tau'_n(\alpha_n) which is estimated by the method described in Section 6 of Padoan and Stupfler (2020). See estExtLevel for details.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for tau'_n-th expectile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting formula in Corollary 4.2, Theorem 6.2 of Padoan and Stupfler (2020) and (46) in Drees (2003). See Sections 4-6 in Padoan and Stupfler (2020) for details. When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.

Value

A list with elements:

• HatQMES: an estimate of the \tau'_n-th quantile based MES;

• VarHatQMES: an estimate of the asymptotic variance of the quantile based MES estimator;

• CIHatQMES: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau'_n-th MES.

Author(s)

Simone Padoan, [e-mail], http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, [e-mail], 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 \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

ExpectMES, HTailIndex, predExpectiles, extQuantile

Examples

# Marginl Expected Shortfall quantile 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
tau1 <- 0.9995

# 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 <- QuantMES(data, NULL, tau1, var=TRUE, k=150, bigBlock=bigBlock,
                   smallBlock=smallBlock)
MESHat