HTailIndex {ExtremeRisks}R Documentation

Hill Tail Index Estimation

Description

Computes a point and interval estimate of the tail index based on the Hill's estimator.

Usage

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

Arguments

data

A vector of (1 x n) observations.

k

An integer specifying the value of the intermediate sequence k_n. See Details.

var

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

varType

A string specifying the 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 biast=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.

alpha

A real in (0,1) specifying the confidence level (1-α)100\% of the approximate confidence interval for the tail index.

Details

For a dataset data of sample size n, the tail index γ of its (marginal) distribution is computed by applying the Hill estimator. The observations can be either independent or temporal dependent.

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.

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.

de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer-Verlag, New York.

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

MLTailIndex, MomTailIndex, EBTailIndex

Examples

# Tail index estimation based on the Hill estimator 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

# Number of larger order statistics
k <- 150

# sample size
ndata <- 2500

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

# tail index estimation
gammaHat1 <- HTailIndex(data, k, TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
gammaHat1$gammaHat
gammaHat1$CIgamHat

# tail index estimation with bias correction
gammaHat2 <- HTailIndex(data, 2*k, TRUE, bias=TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
gammaHat2$gammaHat-gammaHat2$BiasGamHat
gammaHat2$CIgamHat

[Package ExtremeRisks version 0.0.4 Index]