Proportional incremental cost probability functions and their frontiers

The econometric analysis of cost functions is based on the analysis of the conditional distribution of the cost Y given the level of the outputs \(X\in {\mathbb {R}}_+^p\) and given a set of environmental variables \(Z\in {\mathbb {R}}^d\). The model basically describes the conditional distribution of Y given \(X\ge x\) and \(Z=z\). In many applications, the dimension of Z is naturally large and a fully nonparametric specification of the model is limited by the curse of the dimensionality. Most of the approaches so far are based on two-stage estimations when the frontier level does not depend on the value of Z. But even in the case of separability of the frontier, the estimation procedure suffers from several problems, mainly due to the inherent bias of the estimated efficiency scores and the poor rates of convergence of the frontier estimates. In this paper we suggest an alternative semi-parametric model which avoids the drawbacks of the two-stage methods. It is based on a class of model called the Proportional Incremental Cost Functions (PICF), adapted to our setup from the Cox proportional hazard models extensively used in survival analysis for durations models. We define the PICF model, then we examine its properties and propose a semi-parametric estimation. By this way of modeling, we avoid the first stage nonparametric estimation of the frontier and avoid the curse of dimensionality keeping the parametric \(\sqrt{n}\) rates of convergence for the parameters of interest. We are also able to derive \(\sqrt{n}\)-consistent estimator of the conditional order-m robust frontiers (which, by contrast to the full frontier, may depend on Z) and we prove the Gaussian asymptotic properties of the resulting estimators. We illustrate the flexibility and the power of the procedure by some simulated examples and also with some real data sets.