HypoTesting {ExtremeRisks}R Documentation

Wald-Type Hypothesis Testing

Description

Wald-type hypothesis tes for testing equality of high or extreme expectiles and quantiles

Usage

HypoTesting(data, tau, tau1=NULL, type="ExpectRisks", level="extreme",
            method="LAWS", bias=FALSE, k=NULL, alpha=0.05)

Arguments

data

A matrix of (n x d) observations.

tau

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

tau1

A real in (0,1) specifying the extreme level τ'_n. See Details.

type

A string specifying the type of test. By default type="ExpectRisks" specifies the test for testing the equality of expectiles. See Details.

level

A string specifying the level of the expectile. This make sense when type="ExpectRisks". By default level="extreme" specifies that the test concerns expectiles at the extreme level. See Details.

method

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

bias

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

k

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

alpha

A real in (0,1) specifying the significance level of the test.

Details

With a dataset data of d-dimensional observations and sample size n, a Wald-type hypothesis testing is performed in order to check whether the is empirical evidence against the null hypothesis. The null hypothesis concerns the equality among the expectiles or quantiles or tail indices of the marginal distributions. The three tests depend on the depends on the estimation of the d-dimensional tail index γ. Here, γ is estimated using the Hill estimation (see MultiHTailIndex for details). The data are regarded as d-dimensional temporal independent observations coming from dependent variables. See Padoan and Stupfler (2020) for details.

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

See Also

MultiHTailIndex, predMultiExpectiles, extMultiQuantile

Examples

# Hypothesis testing on the equality extreme expectiles based on a sample of
# 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 <- 0.95
# Extreme level (or tail probability 1-tau1 of unobserved expectile)
tau1 <- 0.9995

# 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])

# Performs Wald-type hypothesis testing
HypoTesting(data, tau, tau1)

# Hypothesis testing on the equality extreme expectiles based on a sample of
# d-dimensional observations simulated from a joint distribution with
# a Clayton copula and different Frechet marginal distributions.

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

# parameter setting
dim <- 3
dep <- 2
scale <- rep(1, dim)
shape <- c(2.1, 3, 4.5)
par <- list(dep=dep, scale=scale, shape=shape, dim=dim)

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.95
# Extreme level (or tail probability 1-tau1 of unobserved expectile)
tau1 <- 0.9995

# 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])

# Performs Wald-type hypothesis testing
HypoTesting(data, tau, tau1)

[Package ExtremeRisks version 0.0.4 Index]