npbr (version 1.8)

pick_est: Local Pickands' frontier estimator

Description

Computes the Pickands type of estimator introduced by Gijbels and Peng (2000).

Usage

pick_est(xtab, ytab, x, h, k, type="one-stage")

Value

Returns a numeric vector with the same length as x.

Arguments

xtab

a numeric vector containing the observed inputs \(x_1,\ldots,x_n\).

ytab

a numeric vector of the same length as xtab containing the observed outputs \(y_1,\ldots,y_n\).

x

a numeric vector of evaluation points in which the estimator is to be computed.

h

determines the bandwidth at which the estimate will be computed.

k

a numeric vector of the same length as x, which determines the thresholds at which the Pickands' estimator will be computed.

type

a character equal to "one-stage" or "two-stage".

Author

Abdelaati Daouia and Thibault Laurent.

Details

The local Pickands' frontier estimator (option type="one-stage"), obtained by applying the well-known approach of Dekkers and de Haan (1989) in conjunction with the transformed sample of \(z^{xh}_i\)'s described in the function loc_max, is defined as $$ z^{xh}_{(n-k)} + \left(z^{xh}_{(n-k)}-z^{xh}_{(n-2k)}\right)\{2^{-\log\frac{z^{xh}_{(n-k)}-z^{xh}_{(n-2k)}}{z^{xh}_{(n-2k)}-z^{xh}_{(n-4k)}}/\log 2}-1\}^{-1}. $$ It is based on three upper order statistics \(z^{xh}_{(n-k)}\), \(z^{xh}_{(n-2k)}\), \(z^{xh}_{(n-4k)}\), and depends on \(h\) (see loc_max) as well as an intermediate sequence \(k=k(x,n)\to\infty\) with \(k/n\to 0\) as \(n\to\infty\). The two smoothing parameters \(h\) and \(k\) have to be fixed in the 4th and 5th arguments of the function.

Also, the user can replace each observation \(y_i\) in the strip of width \(2h\) around \(x\) by the resulting local Pickands', leaving all observations outside the strip unchanged. Then, one may apply the DEA estimator (see the function dea_est) to the obtained transformed data, giving the local DEA estimator (option type="two-stage").

References

Dekkers, A.L.M. and L. de Haan (1989). On the estimation of extreme-value index and large quantiles estimation, Annals of Statistics, 17, 1795-1832.

Gijbels, I. and Peng, L. (2000). Estimation of a support curve via order statistics, Extremes, 3, 251-277.

See Also

dea_est

Examples

if (FALSE) {
data("green")
plot(log(OUTPUT)~log(COST), data=green)
x <- seq(min(log(green$COST)), max(log(green$COST)), length.out=101)
h=0.5
nx<-unlist(lapply(x,function(y) length(which(abs(log(green$COST)-y)<=h))))
k<-trunc(nx^0.1)
lines(x, pick_est(log(green$COST), log(green$OUTPUT), x, h=h, k=k), lty=1, col="red")
}