Consider a standard multivariate, cross-sectional model in production economics:

\begin{align} y_i & = f(\boldsymbol{x}_i) + \varepsilon_i \\ & = f(\boldsymbol{x}_i) + v_i - u_i \quad \forall i \notag \end{align}

where \(y_i\) is the output of unit \(i\), \(f: R_+^m \rightarrow R_+\) is the production function (cost function) that characterizes the production technology (cost technology), and \(\boldsymbol{x}_i = (x_{i1}, x_{i2}, \cdots, x_{im})^{'}\) denotes the input vector of unit \(i\). Similar to the literature in Stochastic Frontier analysis (SFA), the presented composite error term \(\varepsilon_i = v_i - u_i\) consists of the inefficiency term \(u_i>0\) and stochastic noise term \(v_i\). To estimate the function \(f\), one could use the parametric and nonparametric methods or neoclassical and frontier models, of which methods are classified based on the specification of \(f\) and error term \(\varepsilon\) (see Kuosmanen and Johnson, 2010). In this paper, we assume certain axiomatic properties (e.g., monotonicity, concavity) instead of \(\textit{a priori}\) functional form for the function \(f\) and apply the nonparametric methods to estimate the function \(f\).

Convex Nonparametric Least Square

Convex Quantile and Expectile Approaches

Contextual Variables

Multiple Outputs (DDF Formulation)

Monotonic Models

Stochastic Nonparametric Envelopment of Data

CNLS-G Algorithm (for large sample)

Plot of estimated function

Data Envelopment Analysis

Free Disposal Hull