npbr (version 1.8)

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, control = list("tm_limit" = 700))

Value

Returns a numeric vector with the same length as x. Returns a vector of NA if no solution has been found by the solver (GLPK).

Arguments

xtab

a numeric vector containing the observed inputs $x_1,\dots ,x_n$.

ytab

a numeric vector of the same length as xtab containing the observed outputs $y_1,\dots ,y_n$.

x

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

deg

an integer (polynomial degree).

control

a list of parameters to the GLPK solver. See *Details* of help(Rglpk_solve_LP).

Author

Hohsuk Noh.

Details

The data edge is modeled by a single polynomial ${\phi }_{\theta }(x)={\theta }_0+{\theta }_{1}x+\cdots +{\theta }_p{x}^p$ 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 $x_1,\dots ,x_n$. The implemented function is the estimate ${\hat{\phi }}_{n,p}(x)={\hat{\theta }}_0+{\hat{\theta }}_{1}x+\cdots +{\hat{\theta }}_p{x}^p$ of $\phi (x)$, where $\hat{\theta }=({\hat{\theta }}_0,{\hat{\theta }}_1,\cdots ,{\hat{\theta }}_p{)}^T$ minimizes ${\int }_a^b{\phi }_{\theta }(x)dx$ over $\theta \in {\text{R}}^{p+1}$ subject to the envelopment constraints ${\phi }_{\theta }(x_i)\ge y_i$, $i=1,\dots ,n$.

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)