Regularization of nonparametric frontier estimators

In production theory and efficiency analysis, we estimate the production frontier, the locus of the maximal attainable level of an output (the production), given a set of inputs (the production factors). In other setups, we estimate rather an input (or cost) frontier, the minimal level of the input (cost) attainable for a given set of outputs (goods or services produced). In both cases the problem can be viewed as estimating a surface under shape constraints (monotonicity, …). In this paper we derive the theory of an estimator of the frontier having an asymptotic normal distribution. It is based on the order-m$m$ partial frontier where we let the order m$m$ to converge to infinity when n→∞$n\to \infty$ but at a slow rate. The final estimator is then corrected for its inherent bias. We thus can view our estimator as a regularized frontier. In addition, the estimator is more robust to extreme values and outliers than the usual nonparametric frontier estimators, like FDH and than the unregularized order-_mn$m_n$ estimator of Cazals et al. (2002) converging to the frontier with a Weibull distribution if _mn→∞$m_n\to \infty$ fast enough when n→∞$n\to \infty$. The performances of our estimators are evaluated in finite samples and compared to other estimators through some Monte-Carlo experiments, showing a better behavior (in terms of robustness, bias, MSE and achieved coverage of the resulting confidence intervals). The practical implementation and the robustness properties are illustrated through simulated data sets but also with a real data set.