MLTailIndex {ExtremeRisks}R Documentation

Maximum Likelihood Tail Index Estimation

Description

Computes a point and interval estimate of the tail index based on the Maximum Likelihood (ML) estimator.

Usage

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

Arguments

data

A vector of (1 \times 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 asymptotic 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.

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-\alpha)100\% of the approximate confidence interval for the tail index.

Details

For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is computed by applying the ML 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.

Drees, H. (2000). Weighted approximations of tail processes for \beta-mixing random variables. Annals of Applied Probability, 10, 1274-1301.

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

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

See Also

HTailIndex, MomTailIndex, EBTailIndex

Examples

# Tail index estimation based on the Maximum Likelihood 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
gammaHat <- MLTailIndex(data, k, TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
gammaHat$gammaHat
gammaHat$CIgamHat

[Package ExtremeRisks version 0.0.4 Index]