Stochastic Data Envelopment Analysis—A review

•Semi-parametric stochastic frontier analysis and chance constrained DEA are compared.•Focus on general tendency in data is compared with focus on individual performance.•A typical management science application is evaluated from the statistical approach.•A typical statistical efficiency application is evaluated from the management science approach.•Approaches to disentangling of random noise and random efficiency are discussed.

This paper provides a review of stochastic Data Envelopment Analysis (DEA). We discuss extensions of deterministic DEA in three directions: (i) deviations from the deterministic frontier are modeled as stochastic variables, (ii) random noise in terms of measurement errors, sample noise, and specification errors is made an integral part of the model, and (iii) the frontier is stochastic as is the underlying Production Possibility Set (PPS).Stochastic DEA utilizes non-parametric convex or conical hull reference technologies based upon axioms from production theory accompanied by a statistical foundation in terms of axioms from statistics or distributional assumptions. The approaches allow for an estimation of stochastic inefficiency compared to a deterministic or a stochastic PPS and for statistical inference while maintaining an axiomatic foundation. Focus is on bridges and differences between approaches within the field of Stochastic DEA including semi-parametric Stochastic Frontier Analysis (SFA) and Chance Constrained DEA (CCDEA).We argue that statistical inference based upon homogenous bootstrapping in contrast to a management science approach imposes a restrictive structure on inefficiency, which may not facilitate the communication of results of the analysis to decision makers. Semi-parametric SFA and CCDEA differ w.r.t. the modeling of noise and stochastic inefficiency. The two approaches are in spite of the inherent differences shown to be complements in the sense that the stochastic PPSs obtained by the two approaches share basic similarities in the case of one output and multiple inputs. Recent contributions related to (i) disentangling of random noise and random inefficiency and (ii) obtaining smooth shape constrained estimators of the frontier are discussed.