loc_max: Local maximum frontier estimators

In npbr: Nonparametric Boundary Regression

Local maximum frontier estimators

Description

Computes the local constant and local DEA boundary estimates proposed by Gijbels and Peng (2000).

Usage

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

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

Details

When estimating \varphi(x), for a given point x\in\R, the methodology of Gijbels and Peng consists of considering a strip around x of width 2h, where h=h_n\to 0 with nh_n\to\infty as n\to\infty, and focusing then on the y_i observations falling into this strip. More precisely, they consider the transformend variables z^{xh}_i = y_i\mathbf{1}_{(|x_i-x|\leq h)}, i=1,\ldots,n, and the corresponding order statistics z^{xh}_{(1)}\le\cdots\le z^{xh}_{(n)}.

The simple maximum z^{xh}_{(n)}=\max_{i=1,\ldots,n}z^{xh}_i defines then the local constant estimator of the frontier point \varphi(x) [option type="one-stage"]. This opens a way to a two-stage estimation procedure as follows. In a first stage, Gijbels and Peng calculate the maximum z^{xh}_{(n)}. Then, they suggest to replace each observation y_i in the strip of width 2h around x by this maximum, leaving all observations outside the strip unchanged. More precisely, they define \tilde{y}_i= y_i if |x_i-x| > h and \tilde{y}_i= z^{xh}_{(n)} if |x_i-x| \leq h either. Then, they apply the DEA estimator (see the function dea_est) to these transformed data (x_i,\tilde{y}_i), giving the local DEA estimator (option type="two-stage"). An ad hoc way of selecting h is by using for instance the function npcdistbw from the np package (see Daouia et al. (2016) for details).

Value

Returns a numeric vector with the same length as x.

Author(s)

Abdelaati Daouia and Thibault Laurent.

References

Daouia, A., Laurent, T. and Noh, H. (2017). npbr: A Package for Nonparametric Boundary Regression in R. Journal of Statistical Software, 79(9), 1-43. doi:10.18637/jss.v079.i09.

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

See Also

dea_est

Examples

data("green")
x.green <- seq(min(log(green$COST)), max(log(green$COST)), 
 length.out=101)
# Local maximum frontier estimates
# a. Local constant estimator
loc_max_1stage<-loc_max(log(green$COST), log(green$OUTPUT), 
 x.green, h=0.5, type="one-stage")
# b. Local DEA estimator
loc_max_2stage<-loc_max(log(green$COST), log(green$OUTPUT), 
 x.green, h=0.5, type="two-stage")  
# Representation 
plot(log(OUTPUT)~log(COST), data=green)
lines(x.green, loc_max_1stage, lty=1, col="magenta")
lines(x.green, loc_max_2stage, lty=2, col="cyan")
legend("topleft",legend=c("one-stage", "two-stage"), 
 col=c("magenta","cyan"), lty=c(1,2))

npbr documentation built on March 31, 2023, 7:45 p.m.