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).

1 2 | ```
ThreeStage(y, x, xstar, tau.e, grid.lam = seq(-2, 2, 0.1), grid.k, tau.lam,
a = 0, tol = 1e-04)
``` |

`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 |

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

Wang, H. and Li, D. (2013). Estimation of conditional high quantiles through power transformation. Journal of the American Statistical Association, 108, 1062-1074.

`TwoStage`

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
#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)
``` |

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