Conical FDH Estimators of Directional Distances and Luenberger Productivity Indices for General Technologies

In productivity and efficiency analysis, directional distances are very popular, due to their flexibility for choosing the direction to evaluate the distance of Decision Making Units (DMUs) to the efficient frontier of the production set. The theory and the statistical properties of these measures are today well known in various situations. But so far, the way to measure directional distances to the cone spanned by the attainable set has not been analyzed. In this paper we fill this gap and describe how to define and estimate directional distances to this cone, for general technologies, i.e. without imposing convexity. Their statistical properties are also developed. This allows us to measure distances to non-convex attainable set under Constant Returns to Scale (CRS) but also to measure and estimate Luenberger productivity indices and their decompositions for general technologies. The way to make inference on these indices is also described in details. We propose illustrations with some simulated data, as well as, a practical example of inference on Luenberger productivity indices and their decompositions with a well-known real data set.