npbr (version 1.8)

loc_max: 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")

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.

type

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

Author

Abdelaati Daouia and Thibault Laurent.

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).

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))