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).

```
kern_smooth(xtab, ytab, x, h, method="u", technique="noh",
control = list("tm_limit" = 700))
```

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).

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).

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 \(i\)th 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 \(s\)th 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 \(\int A_i(x) dx\).

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.

# 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) # }