npbr (version 1.0)

loc_est: Local linear frontier estimator

Description

Computes the local linear smoothing frontier estimator of Hall, Park and Stern (1998).

Usage

loc_est(xtab, ytab, x, h)

Arguments

xtab

numeric vectors containing the observed inputs $x_1,\ldots,x_n$

ytab

numeric vectors 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 local linear estimate will be computed

Value

• Returns a numeric vector with the same length as x

Details

The implemented local linear estimator of $\varphi(x)$ is defined by $$\hat \varphi_{n,LL}(x) = \min \Big{ z : {\rm there~exists~} \theta\geq 0 ~{\rm such~that~} y_i \leq z + \theta(x_i - x)$$ $${\rm for~all}~i~{\rm such~that~}x_i \in (x-h,x+h) \Big}.$$ Hall and Park (2004) proposed a bootstrap procedure for selecting the optimal bandwidth $h$ in $\hat \varphi_{n,LL}$. See the function loc_est_bw.

References

Hall, P. and Park, B.U. (2004). Bandwidth choice for local polynomial estimation of smooth boundaries. Journal of Multivariate Analysis, 91 (2), 240-261. Hall, P., Park, B.U. and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries. Journal of Multivariate Analysis, 66, 71-98.

See Also

poly_est, loc_est_bw

Examples

data("nuclear")
plot(ytab~xtab, data=nuclear)
x <- seq(min(nuclear$xtab), max(nuclear$xtab), length.out=1001)
lines(x, loc_est(nuclear$xtab, nuclear$ytab, x, h=79.12), lty=1, col="red")   
lines(x, loc_est(nuclear$xtab, nuclear$ytab, x, h=40), lty=2, col="blue") 
legend("topleft",legend=c("h=72.70", "h=40"), col=c("red","blue"), 
lty=c(1,2))