MultiHTailIndex {ExtremeRisks}R Documentation

Multidimensional Hill Tail Index Estimation

Description

Computes point estimates and (1α)100%(1-\alpha)100\% confidence regions estimate of dd-dimensional tail indices based on the Hill's estimator.

Usage

MultiHTailIndex(data, k, var=FALSE, varType="asym-Dep", bias=FALSE,
                alpha=0.05, plot=FALSE)

Arguments

data

A matrix of (n×d)(n \times d) observations.

k

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

var

If var=TRUE then an estimate of the variance-covariance matrix of the tail indices estimators is computed.

varType

A string specifying the asymptotic variance to compute. By default varType="asym-Dep" specifies the variance estimator for dd dependent marginal variables. See Details.

bias

A logical value. By default biast=FALSE specifies that no bias correction is computed. See Details.

alpha

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

plot

A logical value. By default plot=FALSE specifies that no graphical representation of the estimates is provided. See Details.

Details

For a dataset data of (n×d)(n \times d) observations, where dd is the number of variables and nn is the sample size, the tail index γ\gamma of the dd marginal distributions is estimated by applying the Hill estimator. Together with a point estimate a (1α)100%(1-\alpha)100\% confidence region is computed. The data are regarded as d-dimensional temporal independent observations coming from dependent variables.

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). Joint inference on extreme expectiles for multivariate heavy-tailed distributions. arXiv e-prints arXiv:2007.08944, https://arxiv.org/abs/2007.08944.

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics to financial 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.

See Also

HTailIndex, rmdata

Examples

# Tail index estimation based on the multivariate Hill estimator obtained with
# n observations simulated from a d-dimensional random vector with a multivariate
# distribution with equal Frechet margins and a Clayton copula.
library(plot3D)
library(copula)
library(evd)

# distributional setting
copula <- "Clayton"
dist <- "Frechet"

# parameter setting
dep <- 3
dim <- 3
scale <- rep(1, dim)
shape <- rep(3, dim)
par <- list(dep=dep, scale=scale, shape=shape, dim=dim)

# Number of larger order statistics
k <- 150

# sample size
ndata <- 1000

# Simulates a sample from a multivariate distribution with equal Frechet
# marginals distributions and a Clayton copula
data <- rmdata(ndata, dist, copula, par)
scatter3D(data[,1], data[,2], data[,3])

# tail indices estimation
est <- MultiHTailIndex(data, k, TRUE)
est$gammaHat
est$VarCovGHat
# run the following command to see the graphical representation

 est <- MultiHTailIndex(data, k, TRUE, plot=TRUE)


[Package ExtremeRisks version 0.0.4 Index]