predExpectiles {ExtremeRisks}R Documentation

Extreme Expectile Estimation

Description

Computes a point and interval estimate of the expectile at the extreme level (Expectile Prediction).

Usage

predExpectiles(data, tau, tau1, method="LAWS", tailest="Hill", 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.

tailest

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

var

If var=TRUE then an estimate of the asymptotic variance of the expectile 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 expectile is computed. The estimation of the expectile at the extreme level tau1 (τ'_n) is meant to be a prediction beyond the observed sample. 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) or in alternative using the the expectile based estimator (see EBTailIndex). The observations can be either independent or temporal dependent. See Section 3.2 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

HTailIndex, EBTailIndex, estExpectiles, extQuantile

Examples

# Extreme expectile estimation at the extreme level tau1 obtained with
# 1-dimensional data simulated from an AR(1) with univariate
# Student-t distributed 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 <- 0.95
# Extreme level (or tail probability 1-tau1 of unobserved expectile)
tau1 <- 0.9995

# sample size
ndata <- 2500

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

# Extreme expectile estimation
expectHat1 <- predExpectiles(data, tau, tau1, var=TRUE, bigBlock=bigBlock,
	                         smallBlock=smallBlock)
expectHat1$EExpcHat
expectHat1$CIExpct
# Extreme expectile estimation with bias correction
tau <- 0.80
expectHat2 <- predExpectiles(data, tau, tau1, "QB", var=TRUE, bias=TRUE, bigBlock=bigBlock,
							 smallBlock=smallBlock)
expectHat2$EExpcHat
expectHat2$CIExpct

[Package ExtremeRisks version 0.0.4 Index]