estMultiExpectiles {ExtremeRisks}

Multidimensional High Expectile Estimation

Description

Computes point estimates and (1-\alpha)100\% confidence regions for d-dimensional expectiles at the intermediate level.

Usage

estMultiExpectiles(data, tau, method="LAWS", tailest="Hill", var=FALSE,
                   varType="asym-Ind-Adj", k=NULL, alpha=0.05, plot=FALSE)

Arguments

data

A matrix of (n \times d) observations.

tau

A real in (0,1) specifying the intermediate level \tau_n. See Details.

method

A string specifying the method used to estimate the expecile. By default est="LAWS" specifies the use of the direct LAWS estimator. See Details.

tailest

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

var

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

varType

A string specifying the asymptotic variance-covariance matrix to compute. By default varType="asym-Ind-Adj" specifies that the variance-covariance matrix is computed assuming dependent variables and exploiting a suitable adjustment. See Details.

k

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

alpha

A real in (0,1) specifying the confidence level (1-\alpha)100\% of the approximate confidence region for the d-dimensional expecile at the intermedite level.

plot

A logical value. By default plot=FALSE specifies that no graphical representation of the estimates is not provided. See Details.

Details

For a dataset data of d-dimensional observations and sample size n, an estimate of the \tau_n-th d-dimensional is computed. Two estimators are available: the so-called direct Least Asymmetrically Weighted Squares (LAWS) and indirect Quantile-Based (QB). The QB estimator depends on the estimation of the d-dimensional tail index \gamma. Here, \gamma is estimated using the Hill estimator (see MultiHTailIndex). The data are regarded as d-dimensional temporal independent observations coming from dependent variables. See Padoan and Stupfler (2020) for details.

• The so-called intermediate level tau or \tau_n is a sequence of positive reals such that \tau_n \to 1 as n \to \infty. Practically, for each individual marginal distribution \tau_n \in (0,1) is the ratio between N (Numerator) and D (Denominator). Where N is the empirical mean distance of the \tau_n-th expectile from the observations smaller than it, and D is the empirical mean distance of \tau_n-th expectile from all the observations.

• If method='LAWS', then the expectile at the intermediate level \tau_n is estimated applying the direct LAWS estimator. Instead, If method='QB' the indirect QB esimtator is used to estimate the expectile. See Section 2.1 in Padoan and Stupfler (2020) for details.

• When the expectile is estimated by the indirect QB esimtator (method='QB'), an estimate of the d-dimensional tail index \gamma is needed. Here the d-dimensional tail index \gamma is estimated using the d-dimensional Hill estimator (tailest='Hill', see MultiHTailIndex). This is the only available option so far (soon more results will be available).

• k or k_n is the value of the so-called intermediate sequence k_n, n=1,2,\ldots. Its represents a sequence of positive integers such that k_n \to \infty and k_n/n \to 0 as n \to \infty. Practically, for each marginal distribution, when method='LAWS' and tau=NULL, k_n specifies by \tau_n=1-k_n/n the intermediate level of the expectile. Instead, for each marginal distribution, when method='QB', then the value k_n specifies the number of k+1 larger order statistics to be used to estimate \gamma by the Hill estimator and if tau=NULL then it also specifies by \tau_n=1-k_n/n the confidence level \tau_n of the quantile to estimate.

• If var=TRUE then an estimate of the asymptotic variance-covariance matrix of the d-dimensional expecile estimator is computed. If the data are regarded as d-dimensional temporal independent observations coming from dependent variables. Then, the asymptotic variance-covariance matrix is estimated by the formulas in section 3.1 of Padoan and Stupfler (2020). In particular, the variance-covariance matrix is computed exploiting the asymptotic behaviour of the relative explectile estimator appropriately normalized and using a suitable adjustment. This is achieved through varType="asym-Ind-Adj". The data can also be regarded as coded-dimensional temporal independent observations coming from independent variables. In this case the asymptotic variance-covariance matrix is diagonal and is also computed exploiting the formulas in section 3.1 of Padoan and Stupfler (2020). This is achieved through varType="asym-Ind".

• Given a small value \alpha\in (0,1) then an asymptotic confidence region for the \tau_n-th expectile, with approximate nominal confidence level (1-\alpha)100\% is computed. In particular, a "symmetric" confidence regions is computed exploiting the asymptotic behaviour of the relative explectile estimator appropriately normalized. See Sections 3.1 of Padoan and Stupfler (2020) for detailed.

• If plot=TRUE then a graphical representation of the estimates is not provided.

Value

A list with elements:

• ExpctHat: an point estimate of the \tau_n-th d-dimensional expecile;

• biasTerm: an point estimate of the bias term of the estimated expecile;

• VarCovEHat: an estimate of the asymptotic variance of the expectile estimator;

• EstConReg: an estimate of the approximate (1-\alpha)100\% confidence region for \tau_n-th d-dimensional expecile.

Author(s)

Simone Padoan, [e-mail], http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, [e-mail], 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

See Also

MultiHTailIndex, predMultiExpectiles, extMultiQuantile

Examples

# Extreme expectile estimation at the intermediate level tau obtained with
# d-dimensional observations simulated from a joint distribution with
# a Gumbel copula and equal Frechet marginal distributions.
library(plot3D)
library(copula)
library(evd)

# distributional setting
copula <- "Gumbel"
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)

# Intermediate level (or sample tail probability 1-tau)
tau <- .95

# sample size
ndata <- 1000

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

# High d-dimensional expectile (intermediate level) estimation
expectHat <- estMultiExpectiles(data, tau, var=TRUE)
expectHat$ExpctHat
expectHat$VarCovEHat
# run the following command to see the graphical representation

 expectHat <- estMultiExpectiles(data, tau, var=TRUE, plot=TRUE)