npbr (version 1.1)

poly_est: Polynomial frontier estimators

Description

Computes the polynomial-type estimators of frontiers and boundaries proposed by Hall, Park and Stern (1998).

Usage

poly_est(xtab, ytab, x, deg)

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.

deg

an integer (polynomial degree).

Value

• Returns a numeric vector with the same length as x.

Details

The data edge is modeled by a single polynomial $\varphi_{\theta}(x) = \theta_0+\theta_1 x+\cdots+\theta_p x^p$ of known degree $p$ that envelopes the full data and minimizes the area under its graph for $x\in[a,b]$, with $a$ and $b$ being respectively the lower and upper endpoints of the design points $x_1,\ldots,x_n$. The implemented function is the estimate $\hat\varphi_{n,p}(x) = \hat\theta_0+\hat\theta_1 x+\cdots+\hat\theta_p x^p$ of $\varphi(x)$, where $\hat\theta=(\hat\theta_0,\hat\theta_1,\cdots,\hat\theta_p)^T$ minimizes $\int_{a}^b \varphi_{\theta}(x) \,dx$ over $\theta\in\R^{p+1}$ subject to the envelopment constraints $\varphi_{\theta}(x_i)\geq y_i$, $i=1,\ldots,n$.

References

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

loc_est

Examples

data("air")
x.air <- seq(min(air$xtab), max(air$xtab), length.out=101)
# Optimal polynomial degrees via the AIC criterion
(p.aic.air<-poly_degree(air$xtab, air$ytab, 
 type = "AIC"))
# Polynomial boundaries estimate 
y.poly.air<-poly_est(air$xtab, air$ytab, x.air, 
 deg=p.aic.air)
# Representation
plot(x.air, y.poly.air, lty=1, lwd=4, 
 col="magenta", type="l")
points(ytab~xtab, data=air)  
legend("topleft",legend=paste("degree =",p.aic.air), 
 col="magenta", lwd=4, lty=1)