estExpectiles: High Expectile Estimation
In ExtremeRisks: Extreme Risk Measures

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/UniEstimation.r View source: R/Estimation.r

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

1 2	estExpectiles(data, tau, method="LAWS", tailest="Hill", var=FALSE, varType="asym-Dep-Adj", bigBlock=NULL, smallBlock=NULL, k=NULL, alpha=0.05)

`data`	A vector of (1 x n) observations.
`tau`	A real in (0,1) specifying the intermediate 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 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-α)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 τ_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 γ. Here, γ 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 τ_n is a sequence of positive reals such that τ_n -> 1 as n -> ∞. Practically, τ_n in (0,1) is the ratio between N (Numerator) and D (Denominator). Where N is the empirical mean distance of the τ_n-th expectile from the observations smaller than it, and D is the empirical mean distance of τ_n-th expectile from all the observations.
If method='LAWS', then the expectile at the intermediate level τ_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 γ is needed. If tailest='Hill' then γ is estimated using the Hill estimator (see also HTailIndex). If tailest='ExpBased' then γ 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,.... Its represents a sequence of positive integers such that k_n -> ∞ and k_n/n -> 0 as n -> ∞. 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 γ by the Hill estimator and if tau=NULL then it also specifies by τ_n=1-k_n/n the confidence level τ_n of the quantile to estimate. Finally, if tailest="ExpBased" and tau=NULL then it also specifies by τ_n=1-k_n/n the intermediate level expectile based esitmator of γ (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 α\in (0,1) then an asymptotic confidence interval for the τ_n-th expectile, with approximate nominal confidence level (1-α)100\% is computed. See Sections 3.1 and C.1 in the Appendix of Padoan and Stupfler (2020).

A list with elements:

ExpctHat: an estimate of the τ_n-th expecile;
VarExpHat: an estimate of the asymptotic variance of the expectile estimator;
CIExpct: an estimate of the approximate (1-α)100\% confidence interval for τ_n-th expecile.

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, http://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.

HTailIndex, EBTailIndex, predExpectiles, extQuantile

# Extreme expectile estimation at the intermediate level tau obtained with
# 1-dimensional data simulated from am 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