| Type: | Package |
| Title: | Extreme Regression of Quantiles |
| Version: | 1.0 |
| Date: | 2016-07-06 |
| Author: | Huixia Judy Wang |
| Maintainer: | Huixia Judy Wang <judywang@gwu.edu> |
| Description: | Estimation for high conditional quantiles based on quantile regression. |
| Depends: | quantreg, mnormt |
| License: | GPL-3 |
| Packaged: | 2016-07-06 15:57:38 UTC; jwang1 |
| NeedsCompilation: | no |
| Repository: | CRAN |
| Date/Publication: | 2016-07-06 23:48:41 |
Hill Estimator of the Extreme Value Index
Description
Hill Estimator of the Extreme Value Index
Usage
EVI.CFG.func(x, tol = 1e-04, min.prop = 0.3, taus)
Arguments
x |
the estimated quantiles at intermediate quantile levels |
tol |
the tolerance level used for checking quantile crossing |
min.prop |
the minimum proportion of quantiles that are estimated higher than the adjacent lower quantiles |
taus |
the corresponding quantile levels |
Details
The function estimates the extreme value index using Hill estimator based on the estimated intermediate quantiles.
Value
The estimated extreme value index is returned. If the proportion of cases with quantile crossing is too high, an NA is returned.
References
Chernozhukov, C., Fernandez-Val, I., and Galichon, A. (2010). Quantile and probability curves without crossing. Econometrica, 78, 1093-1125.
Estimation of the C vector
Description
This function estimates the C vector involved in the function test.EVI for testing the constancy of EVI
Usage
Estc.func(y, x, tau = 0.99, M = 2)
Arguments
y |
a vector of n untransformed responses |
x |
a n x p matrix of n observations and p predictors |
tau |
an upper quantile level close to one |
M |
a constant larger than one that is used for estimating the c vector and thus K(x) function. The default is two |
Value
A p-dimensional vector is returned.
Estimation for Quantile Power Transformation Model
Description
This function estimates the power transformation parameter at a single given quantile level
Usage
PowT.1tau.func(y, x, tau, lams = seq(-2, 2, 0.1), a)
Arguments
y |
a vector of length n representing the response |
x |
a n x p matrix of n observations and p predictors |
tau |
the quantile level of interest |
lams |
a set of transformation parameters for grid search |
a |
the location shift |
Details
This function estimates the transformation parameter lam following the estimation method in Mu and He (2007) such that the conditional quantile of the transformed response is linear in covariates. The transformed response is defined as
\Lambda(y) =
{(y+a)^\lambda-1}\lambda, if \lambda \neq 0; =log(y+a) if \lambda=0.
Value
A list is returned with the following components
lam: the estimated transformation parameter
coef: the estimated quantile coefficient from the power-transformed linear quantile regression
References
Mu, Y. and He, X. (2007). Power transformation toward a linear regression quantile. Journal of the American Statistical Association, 102, 269-279.
Three-Stage Extreme Conditional Quantile Estimator
Description
Provides the estimation of extreme conditional quantile using the three-stage estimation method in Wang and Li (2013). Specifically the function estimates the tau.e-th conditional quantile of Y given x=xstar based on the power-transformed quantile regression model and extreme value theory. The method is based on Hill estimator for the extreme value index and works for heavy-tailed distributions (on the original scale).
Usage
ThreeStage(y, x, xstar, tau.e, grid.lam = seq(-2, 2, 0.1), grid.k, tau.lam,
a = 0, tol = 1e-04)
Arguments
y |
a vector of n responses |
x |
a n x p matrix of n observations and p predictors |
xstar |
a m x p matrix of m observations and p predictors |
tau.e |
the extreme quantile level of interest |
grid.lam |
the set of lambda (transformation parameter) values for grid search |
grid.k |
the grid for the number of upper order statistics involved in Hill estimator; used for searching for the data-adaptive k. If the lenfth of grid.k is 1, then k is fixed at grid.k and no selection is performed. |
tau.lam |
the quantile level used for estimating the transformation parameter |
a |
location shift parameter in the power transformation (introduced to avoid negative y values) |
tol |
the tolerance level for checking quantile crossing issue |
Value
A list is returned with the following components
lam: the estimated power-transformation parameter
k: the selected k, the number of upper order statistics involved in Hill estimator
gamma.x: the estimated x-dependent extreme value index (EVI)
cgmma: the pooled EVI estimation
Q3Stage: the three-stage estimator of the tau.e-th conditional quantile of Y given xstar based on the x-dependent EVI estimation
Q3StageP: the three-stage estimator of the tau.e-th conditional quantile of Y given xstar based on the pooled EVI estimation
References
Wang, H. and Li, D. (2013). Estimation of conditional high quantiles through power transformation. Journal of the American Statistical Association, 108, 1062-1074.
See Also
Examples
#A simulation example (sqrt transformation, heteroscedastic error)
library(EXRQ)
n=500
tau.e = c(0.99, 0.993, 0.995)
set.seed(12368819)
x1 = runif(n, -1, 1)
x2 = runif(n, -1, 1)
sqrty = 2 + x1 + x2 + (1+0.8*x1)*rpareto(n, 0.5)
x = as.matrix(cbind(x1, x2))
y = sqrty^2
xstar = rbind(c(-0.5,0),c(0,-0.5),c(0,0),c(0.5,0),c(0,0.5))
## 3Stage estimator
out.3stage <- ThreeStage(y, x, xstar, tau.e, grid.lam=seq(-0.5, 1.5, 0.1), grid.k=50, tau.lam=0.9)
Two-Stage Extreme Conditional Quantile Estimator
Description
This function provides the Two-Stage estimator in Wang, Li and He (2012) for conditional extreme quantiles based on covariate-dependent extreme value index estimation. The intermediate conditional quantile is estimated by quantile regression of the response on the original scale without any transformation. The method is based on Hill estimator for the extreme value index and works for heavy-tailed distributions.
Usage
TwoStage(y, x, xstar, tau.e, k, tol = 1e-04)
Arguments
y |
a vector of length n representing the response |
x |
a n x p matrix of n observations and p predictors |
xstar |
a m x p matrix of m observations and p predictors representing the covariate of interest |
tau.e |
the extreme quantile level of interest |
k |
the number of upper order statistics used in Hill estimator |
tol |
the tolerance level used for checking quantile crossing |
Value
A list of the following commponents is returned
Q2Stage: the estimated (extrapolated) conditional extreme quantile of the response given x=xstar at the quantile level tau.e
gamma.x: the estimated covariate-dependent extreme value index (Hill estimator associated with x=xstar)
References
Wang, H., Li, D., and He, X. (2012). Estimation of high conditional quantiles for heavytailed distributions, Journal of the American Statistical Association, 107, 1453-1464.
See Also
Examples
#A simulation example (sqrt transformation, heteroscedastic error)
library(EXRQ)
n=500
tau.e = c(0.99, 0.993, 0.995)
set.seed(12368819)
x1 = runif(n, -1, 1)
x2 = runif(n, -1, 1)
sqrty = 2 + x1 + x2 + (1+0.8*x1)*rpareto(n, 0.5)
x = as.matrix(cbind(x1, x2))
y = sqrty^2
xstar = rbind(c(-0.5,0),c(0,-0.5),c(0,0),c(0.5,0),c(0,0.5))
## 2Stage method in Wang, Li and He (2012), no transformation
out.2stage <- TwoStage(y, x, xstar, tau.e, k=50)
Estimation of the Extreme Value Index on the Original Scale
Description
This function estimates the extreme value index on the original scale based on the estimated intermediate conditional quantiles on the transformed scale
Usage
est.gamma.func(taus, Lam.Q, lam, a = 0, tol)
Arguments
taus |
a grid of intermediate high quantile levels |
Lam.Q |
a vector of the same length as taus, representing the estimated intermediate conditional quantiles of Y (at taus) on the transformed scale |
lam |
the power-transformation parameter |
a |
location shift parameter in the power transformation (introduced to avoid negative y values) |
tol |
the tolerance level for checking quantile crossing issue |
Value
A list is returned with the following components.
gamma.x: the estimated EVI. If quantile crossing is too severe, which suggests that the estimated intermediate conditional quantiles are unstable, then NA is returned.
Q: the estimated conditional quantile of Y on the original scale
Quantile of the Pareto Distribution
Description
Quantile of the Pareto Distribution
Usage
qpareto(p, gamma)
Arguments
p |
the quantile level |
gamma |
the shape parameter |
Value
the pth quantile
Random Generation for the Pareto Distribution
Description
Random Generation for the Pareto Distribution
Usage
rpareto(n, gamma)
Arguments
n |
number of observations |
gamma |
the shape parameter |
Value
a vector of n i.i.d. random variables from the Pareto distribution
Selection of the Tuning Parameter k
Description
This function selects the tuning parameter k, the number of upper order statistics involved in Hill estimator of EVI among a grid of points following the method described in Section 3.3 of Wang and Li (2013). The method selects k as the value that minimizes the discrepancy between the estimated x-dependent EVI on the transformed scale and lam times the estimated x-dependent EVI on the original scale
Usage
select.k.func(y, x, Lam.y, lam, a, max.tau, grid.k, n)
Arguments
y |
a vector of n untransformed responses |
x |
a n x p matrix of n observations and p predictors |
Lam.y |
a vector of n power-transformed responses |
lam |
the power-transformation parameter |
a |
location shift parameter in the power transformation (introduced to avoid negative y values) |
max.tau |
the upper bound of the intermediate quantile levels |
grid.k |
the grid for the number of upper order statistics involved in Hill estimator |
n |
the number of observations |
Value
the selected k is returned
References
Wang, H. and Li, D. (2013). Estimation of conditional high quantiles through power transformation. Journal of the American Statistical Association, 108, 1062-1074.
Testing the Constancy of EVI Over Covariates
Description
This function tests whether the extreme value index of Y, gamma(x), is constant or varying across the covariate x by using the test procedure described in Section 3.4 of Wang and Li (2013).
Usage
testC.EVI(y, x, grid.lam = seq(-2, 2, 0.1), grid.k, tau.lam = 0.9,
u.x = 0, a = 0, M = 2, tol = 1e-04)
Arguments
y |
a vector of n untransformed responses |
x |
a n x p matrix of n observations and p predictors |
grid.lam |
a grid of points for power-transformation parameter |
grid.k |
a grid of points for k, the number of upper order statistics involved in Hill estimator |
tau.lam |
the quantile level used for estimating the transformation parameter |
u.x |
the proportion to be trimmed in the x direction |
a |
location shift parameter in the power transformation (introduced to avoid negative y values) |
M |
a constant larger than one that is used for estimating the c vector and thus K(x) function. The default is two |
tol |
the tolerance level for checking quantile crossing issue |
Value
A list is returned with the following components
lam: the estimated power-transformation parameter
k: the selected tuning parameter k, the number of upper order statistics involved in Hill estimator
Tm: the proposed test statistic
scaledTm: the standardized test statistic
pval.iid: the p-value based on iid assumption, that is, assuming that K(x)=1
pval.nid: the p-value based on estimated K(x)=(X'C)^(1/EVI)
gamma.bar: the pooled EVI estimator
hat.gamma: a N-dimensional vector consisting of the estimated x-dependent EVI at x=xstar
xstar: a N x p matrix of N observations and p predictors
References
Wang, H. and Li, D. (2013). Estimation of conditional high quantiles through power transformation. Journal of the American Statistical Association, 108, 1062-1074.
Examples
library(EXRQ)
n=500
tau.e = c(0.99, 0.993, 0.995)
set.seed(12368819)
x1 = runif(n, -1, 1)
x2 = runif(n, -1, 1)
sqrty = 2 + x1 + x2 + (1+0.8*x1)*rpareto(n, 0.5)
x = as.matrix(cbind(x1, x2))
y = sqrty^2
out = testC.EVI(y, x, grid.lam=seq(-0.5, 1.5, 0.1), grid.k=50, tau.lam=0.9)
(Tval = out$scaledTm)
(pval.iid = out$pval.iid)
(pval.nid = out$pval.nid)