eng Science and Education Publishing American Journal of Applied Mathematics and Statistics 2328-7292 2013-07-04 1 1 57 63 10.12691/ajams-1-4-2 AJAMS2013142 article Alexander Y. Vaninsky avaninsky@hostos.cuny.edu 1 Mathematics Department, City University of New York/Hostos Community College, New York, USA The paper introduces stochastic DEA with a Perfect Object (SDEA PO). The Perfect Object (PO) is a virtual Decision Making Unit (DMU) that has the smallest inputs and greatest outputs. Including the PO in a collection of actual objects yields an explicit formula of the efficiency index. Given the distributions of DEA inputs and outputs, this formula allows us to derive the probability distribution of the efficiency score, to find its mathematical expectation, and to deliver common (group¨Crelated) and partial (object-related) efficiency components. We apply this approach to a prospective analysis of environmental efficiency of the major national and regional economies. http://pubs.sciepub.com/ajams/1/4/2/ajams-1-4-2.pdf Data Envelopment Analysis DEA analytical solutions stochastic DEA with a perfect object efficiency decomposition environmental efficiency