poly_est | R Documentation |
Computes the polynomial-type estimators of frontiers and boundaries proposed by Hall, Park and Stern (1998).
poly_est(xtab, ytab, x, deg, control = list("tm_limit" = 700))
xtab |
a numeric vector containing the observed inputs |
ytab |
a numeric vector of the same length as |
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). |
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
.
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).
Hohsuk Noh.
Hall, P., Park, B.U. and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries. Journal of Multivariate Analysis, 66, 71-98.
loc_est
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)
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