Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation

TLDR

The study establishes a formal statistical foundation for data envelopment analysis efficiency evaluation and proposes hypothesis tests derived from its asymptotic distributions. By modeling output deviations as stochastic variables with monotone decreasing densities, the authors demonstrate that DEA estimators are maximum likelihood estimators, are asymptotically consistent, and have asymptotic distributions matching the true inefficiency deviations. The analysis reveals that DEA estimators are maximum likelihood estimators, are asymptotically consistent, exhibit diminishing bias with larger samples, and their asymptotic distribution of inefficiency deviations matches the true distribution.

Abstract

This paper provides a formal statistical basis for the efficiency evaluation techniques of data envelopment analysis (DEA). DEA estimators of the best practice monotone increasing and concave production function are shown to be also maximum likelihood estimators if the deviation of actual output from the efficient output is regarded as a stochastic variable with a monotone decreasing probability density function. While the best practice frontier estimator is biased below the theoretical frontier for a finite sample size, the bias approaches zero for large samples. The DEA estimators exhibit the desirable asymptotic property of consistency, and the asymptotic distribution of the DEA estimators of inefficiency deviations is identical to the true distribution of these deviations. This result is then employed to suggest possible statistical tests of hypotheses based on asymptotic distributions.