Expectile Based Tail Index Estimation

Description

Computes a point estimate of the tail index based on the Expectile Based (EB) estimator.

Usage

EBTailIndex(data, tau, est=NULL)

Arguments

Details

For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is estimated using the EB estimator:

\hat{\gamma}_n^E =\left(1+\frac{\hat{\bar{F}}_n(\tilde{\xi}_{\tau_n})}{1-\tau_n}\right)^{-1} ,

where \hat{\bar{F}}_n is the empirical survival function of the observations, \tilde{\xi}_{\tau_n} is an estimate of the \tau_n-th expectile. The observations can be either independent or temporal dependent. See Padoan and Stupfler (2020) and Daouia et al. (2018) 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, \tau_n \in (0,1) is the ratio between the empirical mean distance of the \tau_n-th expectile from the smaller observations and the empirical mean distance of of the \tau_n-th expectile from all the observations. An estimate of \tau_n-th expectile is computed and used in turn to estimate \gamma.

• The value est, if provided, is meant to be an esitmate of the \tau_n-th expectile which is used to estimate \gamma. On the contrary, if est=NULL, then the routine EBTailIndex estimate first the \tau_n-th expectile expectile and then use it to estimate \gamma.

Value

An estimate of the tain index \gamma.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

See Also

HTailIndex, MomTailIndex, MLTailIndex,

Examples

# Tail index estimation based on the Expectile based estimator obtained with data
# simulated from an AR(1) with 1-dimensional 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

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.97

# 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 <- EBTailIndex(data, tau)
gammaHat

Marginal Expected Shortfall Expectile Based Estimation

Description

Computes a point and interval estimate of the Marginal Expected Shortfall (MES) using an expectile based approach.

Usage

ExpectMES(data, tau, tau1, method="LAWS", var=FALSE, varType="asym-Dep", bias=FALSE,
          bigBlock=NULL, smallBlock=NULL, k=NULL, alpha_n=NULL, alpha=0.05)

Arguments

Details

For a dataset data of sample size n, an estimate of the \tau'_n-th MES is computed. The estimation of the MES at the extreme level tau1 (\tau'_n) is indeed meant to be a prediction. Two estimators are available: the so-called Least Asymmetrically Weighted Squares (LAWS) based estimator and the Quantile-Based (QB) estimator. The definition of both estimators depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex for details). The observations can be either independent or temporal dependent. See Section 4 in 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. See predExpectiles for details.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. See predExpectiles for details.

• When method='LAWS', then the \tau'_n-th MES is estimated using the LAWS based estimator. When method='QB', the expectile is instead estimated using the QB esimtator. See Sectino 4 in Padoan and Stupfler (2020) and in particular Corollary 4.3 and 4.4 for details. The definition of both estimators depend on the estimation of the tail index \gamma. In particular, the tail index \gamma is estimated using the Hill estimator (see HTailIndex).

• If var=TRUE then an esitmate of the asymptotic variance of the tau'_n-th MES is computed. Notice that the estimation of the asymptotic variance is only available when \gamma is estimated using the Hill estimator (see HTailIndex). With independent observations the asymptotic variance is estimated by \hat{\gamma}^2, see Corollary 4.3 in Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Corollary 4.3 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". See Section 4 adn 5 in Padoan and Stupfler (2020) for details. In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• If bias=TRUE then \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance is estimated by the formula in Corollary 3.8, with serial dependent observations, and \hat{\gamma}^2 with independent observation (see e.g. de Drees 2000, for the details).

• 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, when tau=NULL and method='LAWS', then \tau_n=1-k_n/n is the intermediate level of the expectile to be stimated. k_n also specifies the number of k+1 larger order statistics used in the definition of the Hill estimator (see HTailIndex for detail). Differently, When tau=NULL and method='QB', then \tau_n=1-k_n/n is the intermediate level of the quantile to be stimated.

• If the quantile's extreme level is provided by alpha_n, then expectile's extreme level tau'_n is replaced by tau'_n(\alpha_n) which is estimated by the method described in Section 6 of Padoan and Stupfler (2020). See estExtLevel for details.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for tau'_n-th expectile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting formula in Corollary 4.3, 4.4 and Theorem 6.2 of Padoan and Stupfler (2020) and (46) in Drees (2003). See Sections 4-6 in Padoan and Stupfler (2020) for details. When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.

Value

A list with elements:

• HatXMES: an estimate of the \tau'_n-th expectile based MES;

• VarHatXMES: an estimate of the asymptotic variance of the expectile based MES estimator;

• CIHatXMES: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau'_n-th MES.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics tonancial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.

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

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

See Also

QuantMES, HTailIndex, predExpectiles, extQuantile

Examples

# Marginl Expected Shortfall expectile based estimation at the extreme level
# obtained with 2-dimensional data simulated from an AR(1) with bivariate
# Student-t distributed innovations

tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"

# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=df)

# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15

# quantile's extreme level
alpha_n <- 0.999

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)


# Extreme MES expectile based estimation
MESHat <- ExpectMES(data, NULL, NULL, var=TRUE, k=150, bigBlock=bigBlock,
                    smallBlock=smallBlock, alpha_n=alpha_n)
MESHat

Hill Tail Index Estimation

Description

Computes a point and interval estimate of the tail index based on the Hill's estimator.

Usage

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

Arguments

Details

For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is computed by applying the Hill estimator. The observations can be either independent or temporal dependent.

• 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, the value k_n specifies the number of k+1 larger order statistics to be used to estimate \gamma.

• If var=TRUE then an estimate of the asymptotic variance of the Hill estimator is computed. With independent observations the asymptotic variance is estimated by the formula \hat{\gamma}^2, see Theorem 3.2.5 of de Haan and Ferreira (2006). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in 1288 in Drees (2000). This is achieved through varType="asym-Dep". In this latter case the serial dependence is estimated by exploiting the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see Leadbetter et al. (1986). See also formula (11) in Drees (2003). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• If bias=TRUE then an estimate of the bias term of the Hill estimator is computed implementing using formula (4.2) in de Haan et al. (2016). However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.1. Instead for simplicity standard formulas have been used (see de Haan and Ferreira 2006 Theorem 3.2.5 and Drees 2000 page 1288).

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for \gamma, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting the formulas in de Haan and Ferreira (2006) Theorem 3.2.5 and Drees (2000) page 1288. When biast=TRUE the confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.

Value

A list with elements:

• gammaHat: an estimate of tail index \gamma;

• VarGamHat: an estimate of the asymptotic variance of the Hill estimator;

• BiasGamHat: an estimate of bias term of the Hill estimator;

• AdjExtQHat: the adjustment to correct the Weissman estimator of an extreme quantile.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

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

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

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

See Also

MLTailIndex, MomTailIndex, EBTailIndex

Examples

# Tail index estimation based on the Hill 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
gammaHat1 <- HTailIndex(data, k, TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
gammaHat1$gammaHat
gammaHat1$CIgamHat

# tail index estimation with bias correction
gammaHat2 <- HTailIndex(data, 2*k, TRUE, bias=TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
gammaHat2$gammaHat-gammaHat2$BiasGamHat
gammaHat2$CIgamHat

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

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 \gamma. Here, \gamma 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.

• 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 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.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. For each marginal distribution, the value (1-tau'_n) \in (0,1) is meant to be a small tail probability such that (1-\tau'_n)=1/n or (1-\tau'_n) < 1/n. It is also assumed that n(1-\tau'_n) \to C as n \to \infty, where C is a positive finite constant. Typically, C \in (0,1) so it is expected that there are no observations in a data sample that are greater than the expectile at the extreme level \tau_n'.

• When type="ExpectRisks", the null hypothesis of the hypothesis testing concerns the equality among the expectiles of the marginal distributions. See Section 3.3 of Padoan and Stupfler (2020) for details. When type="QuantRisks", the null hypothesis of the hypothesis testing concerns the equality among the quantiles of the marginal distributions. See Section 5 of Padoan and Stupfler (2020) for details. Note that in this case the test is based on the asymptotic distribution of normalized quantile estimator in the logarithmic scale. When type="tails", the null hypothesis of the hypothesis testing concerns the equality among the tail indices of the marginal distributions. See Sections 3.2 and 3.3 of Padoan and Stupfler (2020) for details.

• When type="ExpectRisks", the null hypothesis concerns the equality among the expectiles of the marginal distributions at the intermediate level and this is achieved through level=="inter". In this case the test is obtained exploiting the asymptotic distribution of relative expectile appropriately normalised. See Section 2.1, 3.1 and 3.3 of Padoan and Stupfler (2020) for details. Instead, if level=="extreme" the null hypothesis concerns the equality among the expectiles of the marginal distributions at the extreme level.

• When method='LAWS', then the \tau'_n-th d-dimensional expectile is estimated using the LAWS based estimator. When method='QB', the expectile is instead estimated using the QB esimtator. The definition of both estimators depend on the estimation of the d-dimensional tail index \gamma. The d-dimensional tail index \gamma is estimated using the d-dimensional Hill estimator (tailest='Hill'), see MultiHTailIndex). See Section 2.2 in Padoan and Stupfler (2020) for details.

• If bias=TRUE then d-dimensional \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance-covariance matrix is estimated by the formulas Section 3.2 of Padoan and Stupfler (2020).

• 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 tau=NULL and method='LAWS' or method='QB', then \tau_n=1-k_n/n is the intermediate level of the expectile to be stimated. When tailest='Hill', for each marginal distributions, then k_n specifies the number of k+1 larger order statistics used in the definition of the Hill estimator.

• A small value \alpha\in (0,1) specifies the significance level of Wald-type hypothesis testing.

Value

A list with elements:

• logLikR: the observed value of log-likelihood ratio statistic test;

• critVal: the quantile (critical level) of a chi-square distribution with d degrees of freedom and confidence level \alpha.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

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)

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

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.

• 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, the value k_n specifies the numer of k+1 larger order statistics to be used to estimate \gamma.

• If var=TRUE then the asymptotic variance of the Hill estimator is computed. With independent observations the asymptotic variance is estimated by the formula in Theorem 3.4.2 of de Haan and Ferreira (2006). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in 1288 in Drees (2000). This is achieved through varType="asym-Dep". In this latter case the serial dependence is estimated by exploiting the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see Leadbetter et al. (1986). See also formula (11) in Drees (2003). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• Given a small value \alpha\in (0,1) then an asymptotic confidence interval for the tail index, with approximate nominal confidence level (1-\alpha)100\% is computed.

Value

A list with elements:

• gammaHat: an estimate of tail index \gamma;

• VarGamHat: an estimate of the variance of the ML estimator;

• CIgamHat: an estimate of the approximate (1-\alpha)100\% confidence interval for \gamma.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

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

Moment based Tail Index Estimation

Description

Computes a point estimate of the tail index based on the Moment Based (MB) estimator.

Usage

MomTailIndex(data, k)

Arguments

Details

For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is computed by applying the MB estimator. The observations can be either independent or temporal dependent. For details see de Haan and Ferreira (2006).

• 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, the value k_n specifies the number of k+1 larger order statistics to be used to estimate \gamma.

Value

An estimate of the tail index \gamma.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

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

See Also

HTailIndex, MLTailIndex, EBTailIndex

Examples

# Tail index estimation based on the Moment 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 <- MomTailIndex(data, k)
gammaHat

Multidimensional Hill Tail Index Estimation

Description

Computes point estimates and (1-\alpha)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

Details

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

• 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, the value k_n specifies the number of k+1 larger order statistics to be used to estimate each marginal tail index \gamma_j for j=1,\ldots,d.

• If var=TRUE then an estimate of the asymptotic variance-covariance matrix of the multivariate Hill estimator is computed. With independent observations the asymptotic variance-covariance matrix is estimated by the matrix \hat{\Sigma}^{LAWS}_{j,\ell}(\gamma,R)(1,1), see bottom formula in page 14 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep" which means d dependent marginal variables. When varType="asym-Ind" d marginal variables are regarded as independent and the returned variance-covariance matrix \hat{\Sigma}^{LAWS}_{j,\ell}(\gamma,R)(1,1) is a diagonal matrix with only variance terms.

• If bias=TRUE then an estimate of the bias term of the Hill estimator is computed implementing using formula (4.2) in de Haan et al. (2016). In this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.1 but instead for simplicity the formula at the bottom of page 14 in Padoan and Stupfler (2020) is still used.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence region for \gamma_j, for j=1,\ldots,d, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting the asymptotic normality of multivariate Hill estimator appropriately normalized (the logarithmic scale is not used), see Padoan and Stupfler (2020) for details.

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

Value

A list with elements:

• gammaHat: an estimate of the d tail indices \gamma_j, for j=1,\ldots,d;

• VarCovGHat: an estimate of the asymptotic variance-covariance matrix of the multivariate Hill estimator;

• biasTerm: an estimate of bias term of the multivariate Hill estimator;

• EstConReg: an estimate of the (1-\alpha)100\% confidence region.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

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)

Marginal Expected Shortfall Quantile Based Estimation

Description

Computes a point and interval estimate of the Marginal Expected Shortfall (MES) using a quantile based approach.

Usage

QuantMES(data, tau, tau1, var=FALSE, varType="asym-Dep", bias=FALSE, bigBlock=NULL,
         smallBlock=NULL, k=NULL, alpha=0.05)

Arguments

Details

For a dataset data of sample size n, an estimate of the \tau'_n-th MES is computed. The estimation of the MES at the extreme level tau1 (\tau'_n) is indeed meant to be a prediction. Estimates are obtained through the quantile based estimator defined in page 12 of Padoan and Stupfler (2020). Such an estimator depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex for details). The observations can be either independent or temporal dependent. See Section 4 in 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. See predExpectiles for details.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. See predExpectiles for details.

• If var=TRUE then an esitmate of the asymptotic variance of the tau'_n-th MES is computed. Notice that the estimation of the asymptotic variance is only available when \gamma is estimated using the Hill estimator (see HTailIndex). With independent observations the asymptotic variance is estimated by \hat{\gamma}^2, see Corollary 4.3 in Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Corollary 4.2 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". See Section 4 and 5 in Padoan and Stupfler (2020) for details. In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• If bias=TRUE then \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance is estimated by the formula in Corollary 3.8, with serial dependent observations, and \hat{\gamma}^2 with independent observation (see e.g. de Drees 2000, for the details).

• 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. k_n specifies the number of k+1 larger order statistics used in the definition of the Hill estimator (see HTailIndex for details).

• If the quantile's extreme level is provided by alpha_n, then expectile's extreme level tau'_n is replaced by tau'_n(\alpha_n) which is estimated by the method described in Section 6 of Padoan and Stupfler (2020). See estExtLevel for details.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for tau'_n-th expectile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting formula in Corollary 4.2, Theorem 6.2 of Padoan and Stupfler (2020) and (46) in Drees (2003). See Sections 4-6 in Padoan and Stupfler (2020) for details. When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.

Value

A list with elements:

• HatQMES: an estimate of the \tau'_n-th quantile based MES;

• VarHatQMES: an estimate of the asymptotic variance of the quantile based MES estimator;

• CIHatQMES: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau'_n-th MES.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics tonancial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.

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

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

See Also

ExpectMES, HTailIndex, predExpectiles, extQuantile

Examples

# Marginl Expected Shortfall quantile based estimation at the extreme level
# obtained with 2-dimensional data simulated from an AR(1) with bivariate
# Student-t distributed innovations


tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"

# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=df)

# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15

# quantile's extreme level
tau1 <- 0.9995

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)


# Extreme MES expectile based estimation
MESHat <- QuantMES(data, NULL, tau1, var=TRUE, k=150, bigBlock=bigBlock,
                   smallBlock=smallBlock)
MESHat

Negative log-returns of DOW JONES.

Description

Series of negative log-returns of the U.S. stock market index Dow Jones.

Format

A 8784*2 data frame.

Details

From the series of n = 8785 closing prices S_{t}, t=1,2,..., for the Dow Jones stock market index, recorded from January 29, 1985 to December 12, 2019, the series of negative log-returns.

X_{t+1} = - \log(S_{t+1}/S_t), \quad 1\leq t\leq n-1

is available. Hence the dataset (negative log-returns) contains 8784 observations.

High Expectile Estimation

Description

Computes a point and interval estimate of the expectile at the intermediate level.

Usage

estExpectiles(data, tau, method="LAWS", tailest="Hill", var=FALSE, varType="asym-Dep-Adj",
              bigBlock=NULL, smallBlock=NULL, k=NULL, alpha=0.05)

Arguments

Details

For a dataset data of sample size n, an estimate of the \tau_n-th expectile is computed. Two estimators are available: the so-called direct Least Asymmetrically Weighted Squares (LAWS) and indirect Quantile-Based (QB). The definition of the QB estimator depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex) or in alternative using the the expectile based estimator (see EBTailIndex). The observations can be either independent or temporal dependent. See Section 3.1 in 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, \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 3.1 in Padoan and Stupfler (2020) for details.

• When the expectile is estimated by the indirect QB esimtator (method='QB'), an estimate of the tail index \gamma is needed. If tailest='Hill' then \gamma is estimated using the Hill estimator (see also HTailIndex). If tailest='ExpBased' then \gamma is estimated using the expectile based estimator (see EBTailIndex). See Section 3.1 in Padoan and Stupfler (2020) for details.

• 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, when method='LAWS' and tau=NULL, k_n specifies by \tau_n=1-k_n/n the intermediate level of the expectile. Instead, when method='QB', if tailest="Hill" 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. Finally, if tailest="ExpBased" and tau=NULL then it also specifies by \tau_n=1-k_n/n the intermediate level expectile based esitmator of \gamma (see EBTailIndex).

• If var=TRUE then the asymptotic variance of the expecile estimator is computed. With independent observations the asymptotic variance is computed by the formula Theorem 3.1 of Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Theorem 3.1 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". In this latter case the computation of the asymptotic variance is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see Leadbetter et al. (1986). See also Section C.1 in Appendix of Padoan and Stupfler (2020). The size of the big and small blocks are specified by the parameters bigblock and smallblock, respectively. Still with serial dependent observations, If varType="asym-Dep-Adj", then the asymptotic variance is estimated using formula (C.79) in Padoan and Stupfler (2020), see Section C.1 of the Appendix for details.

• Given a small value \alpha\in (0,1) then an asymptotic confidence interval for the \tau_n-th expectile, with approximate nominal confidence level (1-\alpha)100\% is computed. See Sections 3.1 and C.1 in the Appendix of Padoan and Stupfler (2020).

Value

A list with elements:

• ExpctHat: a point estimate of the \tau_n-th expecile;

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

• CIExpct: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau_n-th expecile.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

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

See Also

HTailIndex, EBTailIndex, predExpectiles, extQuantile

Examples

# Extreme expectile estimation at the intermediate level tau obtained with
# 1-dimensional data simulated from an AR(1) with Student-t innovations

tsDist <- "studentT"
tsType <- "AR"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.99

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# High expectile (intermediate level) estimation
expectHat <- estExpectiles(data, tau, var=TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
expectHat$ExpctHat
expectHat$CIExpct

Extreme Level Estimation

Description

Estimates the expectile's extreme level corresponding to a quantile's extreme level.

Usage

estExtLevel(alpha_n, data=NULL, gammaHat=NULL, VarGamHat=NULL, tailest="Hill", k=NULL,
            var=FALSE, varType="asym-Dep", bigBlock=NULL, smallBlock=NULL, alpha=0.05)

Arguments

Details

For a given extreme level \alpha_n for the \alpha_n-th quantile, an estimate of the extreme level \tau_n'(\alpha_n) is computed such that \xi_{\tau_n'(\alpha_n)}=q_{\alpha_n}. The estimator is defined by

\hat{\tau}_n'(\alpha_n) = 1 - (1 - \alpha_n)\frac{\hat{\gamma}_n}{1-\hat{\gamma}_n}

where \hat{\gamma}_n is a consistent estimator of the tail index \gamma. If a value for the parameter gammaHat is given, then such a value is used to compute \hat{\tau}_n'. If gammaHat is NULL and a dataset is provided through the parameter data, then the tail index \gamma is estimated by a suitable estimator \hat{\gamma}_n. See Section 6 in Padoan and Stupfler (2020) for more details.

• If VarGamHat is specified, i.e. the variance of the tail index estimator, then the variance of the extreme level estimator \hat{\tau}_n' is computed by using such value.

• When estimating the tail index, if tailest='Hill' then \gamma is estimated using the Hill estimator (see also HTailIndex). If tailest='ML' then \gamma is estimated using the Maximum Likelihood estimator (see MLTailIndex). If tailest='ExpBased' then \gamma is estimated using the expectile based estimator (see EBTailIndex). If tailest='Moment' then \gamma is estimated using the moment based estimator (see MomTailIndex). See Padoan and Stupfler (2020) for details.

• 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, when tailest="Hill" then the value k_n specifies the number of k+1 larger order statistics to be used to estimate \gamma by the Hill estimator. See MLTailIndex, EBTailIndex and MomTailIndex for the other estimators.

• If var=TRUE then the asymptotic variance of the extreme level estimator is computed by applying the delta method, i.e.

  Var(\tau_n') = Var(\hat{\gamma}_n) * (\alpha_n-1)^2 / (1-\hat{\gamma}_n)^4

  where Var(\hat{\gamma}_n is provided by VarGamHat or is estimated when esitmating the tail index through tailest='Hill' and tailest='ML'. See HTailIndex and MLTailIndex for details on how the variance is computed.

• Given a small value \alpha\in (0,1) then an asymptotic confidence interval for the extreme level, \tau_n'(\alpha_n), with approximate nominal confidence level (1-\alpha)100\% is computed.

Value

A list with elements:

• tauHat: an estimate of the extreme level \tau_n';

• tauVar: an estimate of the asymptotic variance of the extreme level estimator \hat{\tau}_n'(\alpha_n);

• tauCI: an estimate of the approximate (1-\alpha)100\% confidence interval for the extreme level \tau_n'(\alpha_n).

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

See Also

estExpectiles, predExpectiles, extQuantile

Examples

# Extreme level estimation for a given quantile's extreme level alpha_n
# obtained with 1-dimensional data simulated from an AR(1) with Student-t innovations

tsDist <- "studentT"
tsType <- "AR"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15

# quantile's extreme level
alpha_n <- 0.999

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# expectile's extreme level estimation
tau1Hat <- estExtLevel(alpha_n, data, var=TRUE, k=150, bigBlock=bigBlock,
                       smallBlock=smallBlock)
tau1Hat

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

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 d-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, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

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)

Expectile Computation

Description

Computes the true expectile for some families of parametric models.

Usage

expectiles(par, tau, tsDist="gPareto", tsType="IID", trueMethod="true",
           estMethod="LAWS", nrep=1e+05, ndata=1e+06, burnin=1e+03)

Arguments

Details

For a parametric family of time series models or a parametric family of distributions (for the case of independent observations) the \tau-th expectile (or expectile of level tau) is computed.

• There are two methods to compute the \tau-th expectile. For the Generalised Pareto and Student-t parametric families of distributions, the analytical epxression of the expectile is available. This is used to compute the \tau-th expectile if the parameter trueMethod="true" is specified. For most of parametric family of distributions or parametric families of time series models the analytical epxression of the expectile is not available. In this case an approximate value of the \tau-th expectile is computed via a Monte Carlo method if the parameter trueMethod=="approx" is specified. In particular, ndata observations from a family of time series models (e.g. tsType="AR" and tsDist="studentT") or a sequence of independent and indentically distributed random variables with common family of distributions (e.g. tsType="IID" and tsDist="gPareto") are simulated nrep times. For each simulation the \tau-th expectile is estimate by the estimation method specified by estMethod. The mean of such estimate provides an approximate value of the \tau-th expectile. The available estimator to esitmate the expecile are the direct LAWS (estMethod="LAWS") and the indirect QB (estMethod="QB"), see estExpectiles for details. The available families of distributions are: Generalised Pareto (tsDist="gPareto"), Student-t (tsDist="studentT") and Frechet (tsDist="Frechet"). The available classes of time series with parametric innovations families of distributions are specified in rtimeseries.

Value

The \tau-th expectile.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

See Also

rtimeseries

Examples

# Derivation of the true tau-th expectile for the Pareto distribution
# via accurate simulation

# parameter value
par <- c(1, 0.3)

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.99

trueExp <- expectiles(par, tau)
trueExp


# tau-th expectile of the AR(1) with Student-t innovations
tsDist <- "studentT"
tsType <- "AR"

# Approximation via Monte Carlo methods
trueMethod <- "approx"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.99

trueExp <- expectiles(par, tau, tsDist, tsType, trueMethod)
trueExp

Multidimensional Value-at-Risk (VaR) or Extreme Quantile (EQ) Estimation

Description

Computes point estimates and (1-\alpha)100\% confidence regions for d-dimensional VaR based on the Weissman estimator.

Usage

extMultiQuantile(data, tau, tau1, var=FALSE, varType="asym-Ind-Log", bias=FALSE,
                 k=NULL, alpha=0.05, plot=FALSE)

Arguments

Details

For a dataset data of d-dimensional observations and sample size n, the VaR or EQ, correspoding to the extreme level tau1, is computed by applying the d-dimensional Weissman estimator. The definition of the Weissman estimator depends on the estimation of the d-dimensional tail index \gamma. Here, \gamma is estimated using the Hill estimation (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 variable, (1-\tau_n)\in (0,1) is a small proportion of observations in the observed data sample that exceed the tau_n-th empirical quantile. Such proportion of observations is used to estimate the individual tau_n-th quantile and tail index \gamma.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. For each variable, the value (1-tau'_n) \in (0,1) is meant to be a small tail probability such that (1-\tau'_n)=1/n or (1-\tau'_n) < 1/n. It is also assumed that n(1-\tau'_n) \to C as n \to \infty, where C is a positive finite constant. The value C is the expected number of exceedances of the individual \tau'_n-th quantile. Typically, C \in (0,1) which means that it is expected that there are no observations in a data sample exceeding the individual quantile of level (1-\tau_n').

• If var=TRUE then an estimate of the asymptotic variance-covariance matrix of the tau'_n-th d-dimensional quantile is computed. The data are regarded as temporal independent observations coming from dependent variables. The asymptotic variance-covariance matrix is estimated exploiting the formula in Section 5 of Padoan and Stupfler (2020). In particular, the variance-covariance matrix is computed exploiting the asymptotic behaviour of the normalized quantile estimator which is expressed in logarithmic scale. This is achieved through varType="asym-Ind-Log". If varType="asym-Ind" then the variance-covariance matrix is computed exploiting the asymptotic behaviour of the d-dimensional relative quantile estimator appropriately normalized (see formula in Section 5 of Padoan and Stupfler (2020)).

• If bias=TRUE then an estimate of each individual \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016), which provides a bias corrected version of the Weissman estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. For simplicity standard the variance-covariance matrix is still computed using formula in Section 5 of Padoan and Stupfler (2020).

• 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, the value k_n specifies the number of k+1 larger order statistics to be used to estimate the individual \tau_n-th empirical quantile and individual tail index \gamma_j for j=1,\ldots,d. The intermediate level \tau_n can be seen defined as \tau_n=1-k_n/n.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence region for tau'_n-th d-dimensional quantile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence regions are computed exploiting the asymptotic behaviour of the normalized quantile estimator in logarithmic scale. This is an "asymmetric" region and it is achieved through varType="asym-Ind-Log". A "symmetric" region is obtained exploiting the asymptotic behaviour of the relative quantile estimator appropriately normalized, see formula in Section 5 of Padoan and Stupfler (2020). This is achieved through varType="asym-Ind".

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

Value

A list with elements:

• ExtQHat: an estimate of the d-dimensional VaR or \tau'_n-th d-dimensional quantile;

• VarCovExQHat: an estimate of the asymptotic variance-covariance of the d-dimensional VaR estimator;

• EstConReg: an estimate of the approximate (1-\alpha)100\% confidence regions for the d-dimensional VaR.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

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

MultiHTailIndex, estMultiExpectiles, predMultiExpectiles

Examples

# Extreme quantile estimation at the extreme level tau1 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 <- 0.95
# Extreme level (or tail probability 1-tau1 of unobserved quantile)
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])

# High d-dimensional expectile (intermediate level) estimation
extQHat <- extMultiQuantile(data, tau, tau1, TRUE)

extQHat$ExtQHat
extQHat$VarCovExQHat
# run the following command to see the graphical representation

 extQHat <- extMultiQuantile(data, tau, tau1, TRUE, plot=TRUE)

Value-at-Risk (VaR) or Extreme Quantile (EQ) Estimation

Description

Computes a point and interval estimate of the VaR based on the Weissman estimator.

Usage

extQuantile(data, tau, tau1, var=FALSE, varType="asym-Dep", bias=FALSE, bigBlock=NULL,
            smallBlock=NULL, k=NULL, alpha=0.05)

Arguments

Details

For a dataset data of sample size n, the VaR or EQ, correspoding to the extreme level tau1, is computed by applying the Weissman estimator. The definition of the Weissman estimator depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex). The observations can be either independent or temporal dependent (see e.g. de Haan and Ferreira 2006; Drees 2003; de Haan et al. 2016 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, (1-\tau_n)\in (0,1) is a small proportion of observations in the observed data sample that exceed the tau_n-th empirical quantile. Such proportion of observations is used to estimate the tau_n-th quantile and \gamma.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. The value (1-tau'_n) \in (0,1) is meant to be a small tail probability such that (1-\tau'_n)=1/n or (1-\tau'_n) < 1/n. It is also assumed that n(1-\tau'_n) \to C as n \to \infty, where C is a positive finite constant. The value C is the expected number of exceedances of the \tau'_n-th quantile. Typically, C \in (0,1) which means that it is expected that there are no observations in a data sample exceeding the quantile of level (1-\tau_n').

• If var=TRUE then an estimate of the asymptotic variance of the tau'_n-th quantile is computed. With independent observations the asymptotic variance is estimated by the formula \hat{\gamma}^2 (see e.g. de Drees 2000, 2003, for details). This is achieved through varType="asym-Ind". With serial dependent data the asymptotic variance is estimated by the formula in 1288 in Drees (2000). This is achieved through varType="asym-Dep". In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively. With serial dependent data the asymptotic variance can also be estimated by formula (32) of Drees (2003). This is achieved through varType="asym-Alt-Dep".

• If bias=TRUE then an estimate of the \tau'_n–th quantile is computed using the formula in page 330 of de Haan et al. (2016), which provides a bias corrected version of the Weissman estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity standard formula in Drees (2000) page 1288 is used.

• 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, the value k_n specifies the number of k+1 larger order statistics to be used to estimate the \tau_n-th empirical quantile and \gamma. The intermediate level \tau_n can be seen defined as \tau_n=1-k_n/n.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for tau'_n-th quantile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting the formulas (33) and (46) of Drees (2003). When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details. Furthermore, in this case with serial dependent data the asymptotic variance is estimated using the formula in Drees (2000) page 1288.

Value

A list with elements:

• ExtQHat: an estimate of the VaR or \tau'_n-th quantile;

• VarExQHat: an estimate of the asymptotic variance of the VaR estimator;

• CIExtQ: an estimate of the approximate (1-\alpha)100\% confidence interval for the VaR.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

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

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

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.

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

See Also

HTailIndex, EBTailIndex, estExpectiles

Examples

# Extreme quantile estimation at the level tau1 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

# Intermediate level (or sample tail probability 1-tau)
tau <- 0.97
# Extreme level (or tail probability 1-tau1 of unobserved quantile)
tau1 <- 0.9995

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# VaR (extreme quantile) estimation
extQHat1 <- extQuantile(data, tau, tau1, TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
extQHat1$ExtQHat
extQHat1$CIExtQ

# VaR (extreme quantile) estimation with bias correction
extQHat2 <- extQuantile(data, tau, tau1, TRUE, bias=TRUE, bigBlock=bigBlock, smallBlock=smallBlock)
extQHat2$ExtQHat
extQHat2$CIExtQ

Extreme Expectile Estimation

Description

Computes a point and interval estimate of the expectile at the extreme level (Expectile Prediction).

Usage

predExpectiles(data, tau, tau1, method="LAWS", tailest="Hill", var=FALSE,
               varType="asym-Dep", bias=FALSE, bigBlock=NULL, smallBlock=NULL,
               k=NULL, alpha_n=NULL, alpha=0.05)

Arguments

Details

For a dataset data of sample size n, an estimate of the \tau'_n-th expectile is computed. The estimation of the expectile at the extreme level tau1 (\tau'_n) is meant to be a prediction beyond the observed sample. Two estimators are available: the so-called Least Asymmetrically Weighted Squares (LAWS) based estimator and the Quantile-Based (QB) estimator. The definition of both estimators depends on the estimation of the tail index \gamma. Here, \gamma is estimated using the Hill estimation (see HTailIndex for details) or in alternative using the the expectile based estimator (see EBTailIndex). The observations can be either independent or temporal dependent. See Section 3.2 in 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, \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.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. The value (1-tau'_n) \in (0,1) is meant to be a small tail probability such that (1-\tau'_n)=1/n or (1-\tau'_n) < 1/n. It is also assumed that n(1-\tau'_n) \to C as n \to \infty, where C is a positive finite constant. Typically, C \in (0,1) so it is expected that there are no observations in a data sample that are greater than the expectile at the extreme level \tau_n'.

• When method='LAWS', then the \tau'_n-th expectile is estimated using the LAWS based estimator. When method='QB', the expectile is instead estimated using the QB esimtator. The definition of both estimators depend on the estimation of the tail index \gamma. When tailest='Hill' then \gamma is estimated using the Hill estimator (see HTailIndex). When tailest='ExpBased', then \gamma is estimated using the expectile based estimator (see EBTailIndex). See Section 3.2 in Padoan and Stupfler (2020) for details.

• If var=TRUE then an esitmate of the asymptotic variance of the tau'_n-th expectile is computed. Notice that the estimation of the asymptotic variance is only available when \gamma is estimated using the Hill estimator (see HTailIndex). With independent observations the asymptotic variance is estimated by \hat{\gamma}^2, see the remark below Theorem 3.5 in Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Throrem 3.5 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". See Section 3.2 in Padoan and Stupfler (2020) for details. In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986). The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.

• If bias=TRUE then \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance is estimated by the formula in Corollary 3.8, with serial dependent observations, and \hat{\gamma}^2 with independent observation (see e.g. de Drees 2000, for the details).

• 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, when tau=NULL and method='LAWS', then \tau_n=1-k_n/n is the intermediate level of the expectile to be stimated. The latter is also used to estimate the tail index when tailest='ExpBased'. Instead, if tailest='Hill', then k_n specifies the number of k+1 larger order statistics used in the definition of the Hill estimator. Differently, When tau=NULL and method='QB', then \tau_n=1-k_n/n is the intermediate level of the quantile to be stimated and of the expectile to be stimated when tailest='ExpBased'. Instead, when tailest='Hill' it is the numer of k+1 larger order statistics used in the definition of the Hill estimator.

• If quantile's extreme level is provided by alpha_n, then expectile's extreme level tau'_n(\alpha_n) is replaced by tau'_n(\alpha_n) which is esitmated using the method described in Section 6 of Padoan and Stupfler (2020). See estExtLevel for details.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence interval for tau'_n-th expectile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence intervals are computed exploiting formula (10) and (11) in Padoan and Stupfler (2020) and (46) in Drees (2003). See Section 5 in Padoan and Stupfler (2020) for details. When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.

Value

A list with elements:

• EExpcHat: an estimate of the \tau'_n-th expecile;

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

• CIExpct: an estimate of the approximate (1-\alpha)100\% confidence interval for \tau'_n-th expecile.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.

de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics tonancial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.

Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.

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

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

See Also

HTailIndex, EBTailIndex, estExpectiles, extQuantile

Examples

# Extreme expectile estimation at the extreme level tau1 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

# 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 <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# Extreme expectile estimation
expectHat1 <- predExpectiles(data, tau, tau1, var=TRUE, bigBlock=bigBlock,
	                         smallBlock=smallBlock)
expectHat1$EExpcHat
expectHat1$CIExpct
# Extreme expectile estimation with bias correction
tau <- 0.80
expectHat2 <- predExpectiles(data, tau, tau1, "QB", var=TRUE, bias=TRUE, bigBlock=bigBlock,
							 smallBlock=smallBlock)
expectHat2$EExpcHat
expectHat2$CIExpct

Multidimensional Extreme Expectile Estimation

Description

Computes point estimates and (1-\alpha)100\% confidence regions for d-dimensional expectile at the extreme level (Expectile Prediction).

Usage

predMultiExpectiles(data, tau, tau1, method="LAWS", tailest="Hill", var=FALSE,
                    varType="asym-Ind-Adj-Log", bias=FALSE, k=NULL, alpha=0.05,
                    plot=FALSE)

Arguments

Details

For a dataset data of d-dimensional observations and sample size n, an estimate of the \tau'_n-th d-dimensional expectile is computed. The estimation of the d-dimensional expectile at the extreme level tau1 (\tau'_n) is meant to be a prediction beyond the observed sample. Two estimators are available: the so-called Least Asymmetrically Weighted Squares (LAWS) based estimator and the Quantile-Based (QB) estimator. The definition of both estimators depends on the estimation of the d-dimensional tail index \gamma. Here, \gamma 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.

• 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 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.

• The so-called extreme level tau1 or \tau'_n is a sequence of positive reals such that \tau'_n \to 1 as n \to \infty. For each marginal distribution, the value (1-tau'_n) \in (0,1) is meant to be a small tail probability such that (1-\tau'_n)=1/n or (1-\tau'_n) < 1/n. It is also assumed that n(1-\tau'_n) \to C as n \to \infty, where C is a positive finite constant. Typically, C \in (0,1) so it is expected that there are no observations in a data sample that are greater than the expectile at the extreme level \tau_n'.

• When method='LAWS', then the \tau'_n-th d-dimensional expectile is estimated using the LAWS based estimator. When method='QB', the expectile is instead estimated using the QB esimtator. The definition of both estimators depend on the estimation of the d-dimensional tail index \gamma. 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). See Section 2.2 in Padoan and Stupfler (2020) for details.

• If var=TRUE then an estimate of the asymptotic variance-covariance matrix of the tau'_n-th d-dimensional expectile is computed. Notice that the estimation of the asymptotic variance-covariance matrix is only available when \gamma is estimated using the Hill estimator (see MultiHTailIndex). The data are regarded as temporal independent observations coming from dependent variables. The asymptotic variance-covariance matrix is estimated exploiting the formulas in Section 3.2 of Padoan and Stupfler (2020). The variance-covariance matrix is computed exploiting the asymptotic behaviour of the normalized expectile estimator which is expressed in logarithmic scale. In addition, a suitable adjustment is considered. This is achieved through varType="asym-Ind-Adj-Log". The data can also be regarded as d-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 3.2 of Padoan and Stupfler (2020). This is achieved through varType="asym-Ind-Log". If varType="asym-Ind-Adj", then the variance-covariance matrix is computed exploiting the asymptotic behaviour of the relative expectile estimator appropriately normalized and exploiting a suitable adjustment. This concerns the case of dependent variables. The case of independent variables is achieved through varType="asym-Ind".

• If bias=TRUE then d-dimensional \gamma is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \tau'_n–th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator. However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance-covariance matrix is estimated by the formulas Section 3.2 of Padoan and Stupfler (2020).

• 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 tau=NULL and method='LAWS' or method='QB', then \tau_n=1-k_n/n is the intermediate level of the expectile to be stimated. When tailest='Hill', for each marginal distributions, then k_n specifies the number of k+1 larger order statistics used in the definition of the Hill estimator.

• Given a small value \alpha\in (0,1) then an estimate of an asymptotic confidence region for tau'_n-th d-dimensional expectile, with approximate nominal confidence level (1-\alpha)100\%, is computed. The confidence regions are computed exploiting the formulas in Section 3.2 of Padoan and Stupfler (2020). If varType="asym-Ind-Adj-Log", then an "asymmetric" confidence regions is computed exploiting the asymptotic behaviour of the normalized expectile estimator in logarithmic scale and using a suitable adjustment. This choice is recommended. If varType="asym-Ind-Adj", then the a "symmetric" confidence regions is computed exploiting the asymptotic behaviour of the relative explectile estimator appropriately normalized.

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

Value

A list with elements:

• ExpctHat: an estimate of the \tau'_n-th d-dimensional expecile;

• biasTerm: an estimate of the bias term of yje \tau'_n-th d-dimensional expecile;

• VarCovEHat: an estimate of the asymptotic variance-covariance of the d-dimensional expectile estimator;

• EstConReg: an estimate of the approximate (1-\alpha)100\% confidence regions for \tau'_n-th d-dimensional expecile.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

See Also

MultiHTailIndex, estMultiExpectiles, extMultiQuantile

Examples

# Extreme expectile estimation at the extreme level tau1 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 <- 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])

# High d-dimensional expectile (intermediate level) estimation
expectHat <- predMultiExpectiles(data, tau, tau1, var=TRUE)

expectHat$ExpctHat
expectHat$VarCovEHat
# run the following command to see the graphical representation

 expectHat <- predMultiExpectiles(data, tau, tau1, var=TRUE, plot=TRUE)

Simulation of Two-Dimensional Temporally Dependent Observations

Description

Simulates samples from parametric families of bivariate time series models.

Usage

rbtimeseries(ndata, dist="studentT", type="AR", copula="Gumbel", par, burnin=1e+03)

Arguments

Details

For a time series class (type), with a parametric family (dist) for the innovations, a sample of size ndata is simulated. See for example Brockwell and Davis (2016).

• The available categories of bivariate time series models are: Auto-Regressive (type="AR"), Auto-Regressive and Moving-Average (type="ARMA"), Generalized-Autoregressive-Conditional-Heteroskedasticity (type="GARCH") and Auto-Regressive.

• With AR(1) times series the available families of distributions for the innovations and the dependence structure (copula) are:

  • Student-t (dist="studentT" and copula="studentT") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) and dependence parameter dep\in(-1,1). The parameters are specified as par <- list(corr, df, dep);

  • Asymmetric Student-t (dist="AStudentT" and copula="studentT") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) and dependence parameter dep\in(-1,1). The paraters are specified as par <- list(corr, df, dep). Note that in this case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details;

• With ARMA(1,1) times series the available families of distributions for the innovations and the dependence structure (copula) are:

  • symmetric Pareto (dist="double-Pareto" and copula="Gumbel" or copula="Gaussian") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), and dependence parameter dep (dep>0 if copula="Gumbel" or dep\in(-1,1) if copula="Gaussian"). The parameters are specified as par <- list(corr, scale, shape, smooth, dep).

  • symmetric Pareto (dist="double-Pareto" and copula="Gumbel" or copula="Gaussian") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), and dependence parameter dep (dep>0 if copula="Gumbel" or dep\in(-1,1) if copula="Gaussian"). The parameters are specified as par <- list(corr, scale, shape, smooth, dep). Note that in this case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details;

• With ARCH(1)/GARCH(1,1) time series the distribution of the innovations are symmetric Gaussian (dist="Gaussian") or asymmetric Gaussian dist="AGaussian". In both cases the marginal parameters (equal for both distributions) are: \alpha_0, \alpha_1, \beta. In the asymmetric Gaussian case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details. The available copulas are: Gaussian (copula="Gaussian") with dependence parameter dep\in(-1,1), Student-t (copula="studentT") with dependence parameters dep\in(-1,1) and \nu>0 (degrees of freedom), Gumbel (copula="Gumbel") with dependence parameter dep>0. The parameters are specified as par <- list(alpha0, alpha1, beta, dep) or par <- list(alpha0, alpha1, beta, dep, df).

Value

A vector of (2 \times n) observations simulated from a specified bivariate time series model.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Brockwell, Peter J., and Richard A. Davis. (2016). Introduction to time series and forecasting. Springer.

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

See Also

rtimeseries, expectiles

Examples

# Data simulation from a 2-dimensional AR(1) with bivariate Student-t distributed
# innovations, with one marginal distribution whose lower and upper tail indices
# that are different

tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"

# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=df)

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)

# Extreme expectile estimation
plot(data, pch=21)
plot(data[,1], type="l")
plot(data[,2], type="l")

Simulation of d-Dimensional Temporally Independent Observations

Description

Simulates samples of independent d-dimensional observations from parametric families of joint distributions with a given copula and equal marginal distributions.

Usage

rmdata (ndata, dist="studentT", copula="studentT", par)

Arguments

Details

For a joint multivariate distribution with a given parametric copula class (copula) and a given parametric family of equal marginal distributions (dist), a sample of size ndata is simulated.

• The available copula classes are: Student-t (copula="studentT") with \nu>0 degrees of freedom (df) and scale parameters \rho_{i,j}\in (-1,1) for i \neq j=1,\ldots,d (sigma), Gaussian (copula="Gaussian") with correlation parameters \rho_{i,j}\in (-1,1) for i \neq j=1,\ldots,d (sigma), Clayton (copula="Clayton") with dependence parameter \theta>0 (dep), Gumbel (copula="Gumbel") with dependence parameter \theta\geq 1 (dep) and Frank (copula="Frank") with dependence parameter \theta>0 (dep).

• The available families of marginal distributions are:

  • Student-t (dist="studentT") with \nu>0 degrees of freedom (df);

  • Asymmetric Student-t (dist="AStudentT") with \nu>0 degrees of freedom (df). In this case all the observations are only positive;

  • Frechet (dist="Frechet") with scale \sigma>0 (scale) and shape \alpha>0 (shape) parameters.

  • Frechet (dist="double-Frechet") with scale \sigma>0 (scale) and shape \alpha>0 (shape) parameters. In this case positive and negative observations are allowed;

  • symmetric Pareto (dist="double-Pareto") with scale \sigma>0 (scale) and shape \alpha>0 (shape) parameters. In this case positive and negative observations are allowed.

• The available classes of multivariate joint distributions are:

  • studentT-studentT (dist="studentT" and copula="studentT") with parameters par <- list(df, sigma);

  • studentT (dist="studentT" and copula="None" with parameters par <- list(df, dim). In this case the d variables are regarded as independent;

  • studentT-AstudentT (dist="AstudentT" and copula="studentT") with parameters par <- list(df, sigma, shape);

  • Gaussian-studentT (dist="studentT" and copula="Gaussian") with parameters par <- list(df, sigma);

  • Gaussian-AstudentT (dist="AstudentT" and copula="Gaussian") with parameters par <- list(df, sigma, shape);

  • Frechet (dist="Frechet" and copula="None") with parameters par <- list(shape, dim). In this case the d variables are regarded as independent;

  • Clayton-Frechet (dist="Frechet" and copula="Clayton") with parameters par <- list(dep, dim, scale, shape);

  • Gumbel-Frechet (dist="Frechet" and copula="Gumbel") with parameters par <- list(dep, dim, scale, shape);

  • Frank-Frechet (dist="Frechet" and copula="Frank") with parameters par <- list(dep, dim, scale, shape);

  • Clayton-double-Frechet (dist="double-Frechet" and copula="Clayton") with parameters par <- list(dep, dim, scale, shape);

  • Gumbel-double-Frechet (dist="double-Frechet" and copula="Gumbel") with parameters par <- list(dep, dim, scale, shape);

  • Frank-double-Frechet (dist="double-Frechet" and copula="Frank") with parameters par <- list(dep, dim, scale, shape);

  • Clayton-double-Pareto (dist="double-Pareto" and copula="Clayton") with parameters par <- list(dep, dim, scale, shape);

  • Gumbel-double-Pareto (dist="double-Pareto" and copula="Gumbel") with parameters par <- list(dep, dim, scale, shape);

  • Frank-double-Pareto (dist="double-Pareto" and copula="Frank") with parameters par <- list(dep, dim, scale, shape).

  Note that above dim indicates the number of d marginal variables.

Value

A matrix of (n \times d) observations simulated from a specified multivariate parametric joint distribution.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall/CRC Press, Boca Raton, USA.

Simone A. Padoan and Gilles Stupfler (2022). Joint inference on extreme expectiles for multivariate heavy-tailed distributions, Bernoulli 28(2), 1021-1048.

See Also

rtimeseries, rbtimeseries

Examples

library(plot3D)
library(copula)
library(evd)

# Data simulation from a 3-dimensional random vector a with multivariate distribution
# given by a Gumbel copula and three equal Frechet marginal distributions

# 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)

# sample size
ndata <- 1000

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


# Data simulation from a 3-dimensional random vector a with multivariate distribution
# given by a Gaussian copula and three equal Student-t marginal distributions

# distributional setting
dist <- "studentT"
copula <- "Gaussian"

# parameter setting
rho <- c(0.9, 0.8, 0.7)
sigma <- c(1, 1, 1)
Sigma <- sigma^2 * diag(dim)
Sigma[lower.tri(Sigma)] <- rho
Sigma <- t(Sigma)
Sigma[lower.tri(Sigma)] <- rho
df <- 3
par <- list(sigma=Sigma, df=df)

# sample size
ndata <- 1000

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

Simulation of One-Dimensional Temporally Dependent Observations

Description

Simulates samples from parametric families of time series models.

Usage

rtimeseries(ndata, dist="studentT", type="AR", par, burnin=1e+03)

Arguments

Details

For a time series class (type) with a parametric family (dist) for the innovations, a sample of size ndata is simulated. See for example Brockwell and Davis (2016).

• The available categories of time series models are: Auto-Regressive (type="AR"), Auto-Regressive and Moving-Average (type="ARMA"), Generalized-Autoregressive-Conditional-Heteroskedasticity (type="GARCH") and Auto-Regressive and Moving-Maxima (type="ARMAX").

• With AR(1) and ARMA(1,1) times series the available families of distributions for the innovations are:

  • Student-t (dist="studentT") with parameters: \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) specified by par=c(corr, df);

  • symmetric Frechet (dist="double-Frechet") with parameters \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), specified by par=c(corr, scale, shape, smooth);

  • symmetric Pareto (dist="double-Pareto") with parameters \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), specified by par=c(corr, scale, shape, smooth).

  With ARCH(1)/GARCH(1,1) time series the Gaussian family of distributions is available for the innovations (dist="Gaussian") with parameters, \alpha_0, \alpha_1, \beta specified by par=c(alpha0, alpha1, beta). Finally, with ARMAX(1) times series the Frechet families of distributions is available for the innovations (dist="Frechet") with parameters, \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape) specified by par=c(corr, scale, shape).

Value

A vector of (1 \times n) observations simulated from a specified time series model.

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/

References

Brockwell, Peter J., and Richard A. Davis. (2016). Introduction to time series and forecasting. Springer.

Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.

See Also

expectiles

Examples

# Data simulation from a 1-dimensional AR(1) with univariate Student-t
# distributed innovations

tsDist <- "studentT"
tsType <- "AR"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# Graphic representation
plot(data, type="l")
acf(data)

Negative log-returns of S&P 500.

Description

Series of negative log-returns of the U.S. stock market index Standard and Poor 500.

Format

A 8784*2 data frame.

Details

From the series of n = 8785 closing prices S_{t}, t=1,2,..., for the Standard and Poor 500 stock market index, recorded from January 29, 1985 to December 12, 2019, the series of negative log-returns.

X_{t+1} = - \log(S_{t+1}/S_t), \quad 1\leq t\leq n-1

is available. Hence the dataset (negative log-returns) contains 8784 observations.