Fast and efficient computation of directional distance estimators

Directional distances provide useful, flexible measures of technical efficiency of production units relative to the efficient frontier of the attainable set in input-output space. In addition, the additive nature of directional distances permits negative input or outputs quantities. The choice of the direction allows analysis of different strategies for the units attempting to reach the efficient frontier. Simar et al. (Eur J Oper Res 220:853–864, 2012 ) and Simar and Vanhems (J Econom 166:342–354, 2012 ) develop asymptotic properties of full-envelopment, FDH and DEA estimators of directional distances as well as robust order- m and order- $$\alpha $$ α directional distance estimators. Extensions of these estimators to measures conditioned on environmental variables Z are also available (e.g., see Daraio and Simar in Eur J Oper Res 237:358–369, 2014 ). The resulting estimators have been shown to share the properties of their corresponding radial measures. However, to date the algorithms proposed for computing the directional distance estimates suffer from various numerical drawbacks (Daraio and Simar in Eur J Oper Res 237:358–369, 2014 ). In particular, for the order- m versions (conditional and unconditional) only approximations, based on Monte-Carlo methods, have been suggested, involving additional computational burden. In this paper we propose a new fast and efficient method to compute exact values of the directional distance estimates for all the cases (full and partial frontier cases, unconditional or conditional to external factors), that overcome all previous difficulties. This new method is illustrated on simulated and real data sets. Matlab code for computation is provided in an “Appendix”.