Three-Stage Extreme Conditional Quantile Estimator

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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

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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

TwoStage

Examples

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#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)