Probabilistic Characterization of Directional Distances

In productivity analysis, the performance of production units is measured through the distance of the individual decision making units (DMU) to the technology which is defined as the frontier of the production set. Most of the existing methods, Farrell-Debreu and Shephard radial measures (input or output oriented) and hyperbolic distance functions, rely on multiplicative measures of the distance. Directional distance function is an alternative that can be viewed as an additive measure of efficiency. It allows to measure the distance to the efficient frontier along any subsets of inputs and outputs. The directional distances can thus be viewed as a generalization of the standard radial distances. We show in this paper that using a probabilistic formulation of the production process, inspired from Cazals et al. (2002), the directional distance can be expressed as simple radial or hyperbolic distance up to a simple transformation of the inputs/outputs space. This allows to propose simple methods of estimation but also to transfer easily most of the known properties of the estimators shared by the radial and hyperbolic distances. In addition, the formulation allows to define robust directional distances in the lines of α-quantile or order-m partial frontiers. Finally we can also define conditional directional distance functions, conditional to environmental factors. To illustrate the methodology, we show how it can be implemented using a Mutual Funds database.