MultiHTailIndex {ExtremeRisks}R Documentation

Multidimensional Hill Tail Index Estimation

Description

Computes point estimates and (1-α)100\% confidence regions estimate of d-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 x d) 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-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 d 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) specifying the confidence level (1-α)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 x d) observations, where d is the number of variables and n is the sample size, the tail index γ of the d marginal distributions is estimated by applying the Hill estimator. Together with a point estimate a (1-α)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]