poly_est {npbr}

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))

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).

control

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

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.

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).

Author(s)

Hohsuk Noh.

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)