kern_smooth: Frontier estimation via kernel smoothing

In npbr: Nonparametric Boundary Regression

Frontier estimation via kernel smoothing

Description

The function kern_smooth implements two frontier estimators based on kernel smoothing techniques. One is from Noh (2014) and the other is from Parmeter and Racine (2013).

Usage

kern_smooth(xtab, ytab, x, h, method="u", technique="noh", 
 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.
h	determines the bandwidth at which the smoothed kernel estimate will be computed.
method	a character equal to "u" (unconstrained estimator), "m" (under the monotonicity constraint) or "mc" (under simultaneous monotonicity and concavity constraints).
technique	which estimation method to use. "Noh"" specifies the use of the method in Noh (2014) and "pr" is for the method in Parameter and Racine (2013).
control	a list of parameters to the GLPK solver. See *Details* of help(Rglpk_solve_LP).

Details

To estimate the frontier function, Parameter and Racine (2013) considered the following generalization of linear regression smoothers

\hat \varphi(x|p) = \sum_{i=1}^n p_i A_i(x)y_i,

where A_i(x) is the kernel weight function of x for the ith data depending on x_i's and the sort of linear smoothers. For example, the Nadaraya-Watson kernel weights are A_i(x) = K_i(x)/(\sum_{j=1}^n K_j(x)), where K_i(x) = h^{-1} K\{ (x-x_i)/h\}, with the kernel function K being a bounded and symmetric probability density, and h is a bandwidth. Then, the weight vector p=(p_1,\ldots,p_n)^T is chosen to minimize the distance D(p)=(p-p_u)^T(p-p_u) subject to the envelopment constraints and the choice of the shape constraints, where p_u is an n-dimensional vector with all elements being one. The envelopement and shape constraints are

\begin{array}{rcl} \hat \varphi(x_i|p) - y_i &=& \sum_{i=1}^n p_i A_i(x_i)y_i - y_i \geq 0,~i=1,\ldots,n;~~~{\sf (envelopment~constraints)} \\ \hat \varphi^{(1)}(x|p) &=& \sum_{i=1}^n p_i A_i^{(1)}(x)y_i \geq 0,~x \in \mathcal{M};~~~{\sf (monotonocity~constraints)} \\ \hat \varphi^{(2)}(x|p) &=& \sum_{i=1}^n p_i A_i^{(2)}(x)y_i \leq 0,~x \in \mathcal{C},~~~{\sf (concavity~constraints)} \end{array}

where \hat \varphi^{(s)}(x|p) = \sum_{i=1}^n p_i A_i^{(s)}(x) y_i is the sth derivative of \hat \varphi(x|p), with \mathcal{M} and \mathcal{C} being the collections of points where monotonicity and concavity are imposed, respectively. In our implementation of the estimator, we simply take the entire dataset \{(x_i,y_i),~i=1,\ldots,n\} to be \mathcal{M} and \mathcal{C} and, in case of small samples, we augment the sample points by an equispaced grid of length 201 over the observed support [\min_i x_i,\max_i x_i] of X. For the weight A_i(x), we use the Nadaraya-Watson weights.

Noh (2014) considered the same generalization of linear smoothers \hat \varphi(x|p) for frontier estimation, but with a difference choice of the weight p. Using the same envelopment and shape constraints as Parmeter and Racine (2013), the weight vector p is chosen to minimize the area under the fitted curve \hat \varphi(x|p), that is A(p) = \int_a^b\hat \varphi(x|p) dx = \sum_{i=1}^n p_i y_i \left( \int_a^b A_i(x) dx \right), where [a,b] is the true support of X. In practice, we integrate over the observed support [\min_i x_i,\max_i x_i] since the theoretic one is unknown. In what concerns the kernel weights A_i(x), we use the Priestley-Chao weights

A_i(x) = \left\{ \begin{array}{cc} 0 &,~i=1 \\ (x_i - x_{i-1}) K_i(x) &,~i \neq 1 \end{array} \right.,

where it is assumed that the pairs (x_i,y_i) have been ordered so that x_1 \leq \cdots \leq x_n. The choice of such weights is motivated by their convenience for the evaluation of the integral \int A_i(x) dx.

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

Noh, H. (2014). Frontier estimation using kernel smoothing estimators with data transformation. Journal of the Korean Statistical Society, 43, 503-512.

Parmeter, C.F. and Racine, J.S. (2013). Smooth constrained frontier analysis in Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis, Springer-Verlag, New York, 463-488.

See Also

kern_smooth_bw.

Examples

## Not run: 
data("green")
x.green <- seq(min(log(green$COST)), max(log(green$COST)), 
 length.out = 101)
options(np.tree=TRUE, crs.messages=FALSE, np.messages=FALSE)
# 1. Unconstrained 
(h.bic.green.u <- kern_smooth_bw(log(green$COST), 
 log(green$OUTPUT), method = "u", technique = "noh", 
 bw_method = "bic"))
y.ks.green.u <- kern_smooth(log(green$COST), 
 log(green$OUTPUT), x.green, h = h.bic.green.u, 
 method = "u", technique = "noh")

# 2. Monotonicity constraint
(h.bic.green.m <- kern_smooth_bw(log(green$COST),
 log(green$OUTPUT), method = "m", technique = "noh",
 bw_method = "bic"))
y.ks.green.m <- kern_smooth(log(green$COST), 
 log(green$OUTPUT), x.green, h = h.bic.green.m, 
 method = "m", technique = "noh")

# 3. Monotonicity and Concavity constraints
(h.bic.green.mc<-kern_smooth_bw(log(green$COST), log(green$OUTPUT), 
 method="mc", technique="noh", bw_method="bic"))
y.ks.green.mc<-kern_smooth(log(green$COST), 
 log(green$OUTPUT), x.green, h=h.bic.green.mc, method="mc", 
 technique="noh")

# Representation 
plot(log(OUTPUT)~log(COST), data=green, xlab="log(COST)", 
 ylab="log(OUTPUT)") 
lines(x.green, y.ks.green.u, lty=1, lwd=4, col="green")
lines(x.green, y.ks.green.m, lty=2, lwd=4, col="cyan")
lines(x.green, y.ks.green.mc, lty=3, lwd=4, col="magenta")   
legend("topleft", col=c("green","cyan","magenta"), 
lty=c(1,2,3), legend=c("unconstrained", "monotone", 
 "monotone + concave"), lwd=4, cex=0.8)

## End(Not run)    

npbr documentation built on March 31, 2023, 7:45 p.m.