estExpectiles {ExtremeRisks}R Documentation

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 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.

Details

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.

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.

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

[Package ExtremeRisks version 0.0.4 Index]