estExpectiles {ExtremeRisks}

High Expectile Estimation

Description

Computes a point and interval estimate of the expectile at the intermediate level.

Usage

estExpectiles(data, tau, method="LAWS", tailest="Hill", var=FALSE, varType="asym-Dep-Adj",
              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.

method

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

tailest

A string specifying the type of tail index estimator. By default tailest="Hill" specifies the use of Hill estimator. See Details.

var

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

varType

A string specifying the asymptotic variance to compute. By default varType="asym-Dep-Adj" specifies the variance estimator for serial dependent observations implemented with a suitable adjustment. 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 expectile is computed. Two estimators are available: the so-called direct Least Asymmetrically Weighted Squares (LAWS) and indirect Quantile-Based (QB). The definition of the QB estimator depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex) or in alternative using the the expectile based estimator (see EBTailIndex). The observations can be either independent or temporal dependent. See Section 3.1 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. Practically, \tau_n \in (0,1) is the ratio between N (Numerator) and D (Denominator). Where N is the empirical mean distance of the \tau_n-th expectile from the observations smaller than it, and D is the empirical mean distance of \tau_n-th expectile from all the observations.

• If method='LAWS', then the expectile at the intermediate level \tau_n is estimated applying the direct LAWS estimator. Instead, If method='QB' the indirect QB esimtator is used to estimate the expectile. See Section 3.1 in Padoan and Stupfler (2020) for details.

• When the expectile is estimated by the indirect QB esimtator (method='QB'), an estimate of the tail index \gamma is needed. If tailest='Hill' then \gamma is estimated using the Hill estimator (see also HTailIndex). If tailest='ExpBased' then \gamma is estimated using the expectile based estimator (see EBTailIndex). See Section 3.1 in Padoan and Stupfler (2020) for 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. Practically, when method='LAWS' and tau=NULL, k_n specifies by \tau_n=1-k_n/n the intermediate level of the expectile. Instead, when method='QB', if 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 and if tau=NULL then it also specifies by \tau_n=1-k_n/n the confidence level \tau_n of the quantile to estimate. Finally, if tailest="ExpBased" and tau=NULL then it also specifies by \tau_n=1-k_n/n the intermediate level expectile based esitmator of \gamma (see EBTailIndex).

• If var=TRUE then the asymptotic variance of the expecile estimator is computed. With independent observations the asymptotic variance is computed by the formula Theorem 3.1 of Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Theorem 3.1 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". In this latter case the computation of the asymptotic variance is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see Leadbetter et al. (1986). See also Section C.1 in Appendix of Padoan and Stupfler (2020). The size of the big and small blocks are specified by the parameters bigblock and smallblock, respectively. Still with serial dependent observations, If varType="asym-Dep-Adj", then the asymptotic variance is estimated using formula (C.79) in Padoan and Stupfler (2020), see Section C.1 of the Appendix for details.

• Given a small value \alpha\in (0,1) then an asymptotic confidence interval for the \tau_n-th expectile, with approximate nominal confidence level (1-\alpha)100\% is computed. See Sections 3.1 and C.1 in the Appendix of Padoan and Stupfler (2020).

Value

A list with elements:

• ExpctHat: a point estimate of the \tau_n-th expecile;

• VarExpHat: an estimate of the asymptotic variance of the expectile estimator;

• CIExpct: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau_n-th expecile.

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.

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

See Also

HTailIndex, EBTailIndex, predExpectiles, extQuantile

Examples

# Extreme expectile estimation at the intermediate level tau 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

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.99

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# High expectile (intermediate level) estimation
expectHat <- estExpectiles(data, tau, var=TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
expectHat$ExpctHat
expectHat$CIExpct