Abstract
Conventional non-parametric linear programming (LP) based data envelopment analysis (DEA) models have the advantage of being able to estimate multiple input-output efficiency metrics but suffer from sensitivity to outliers and statistical observational noise. Previous observation-deleting approaches to the outlier/noise problem have been somewhat ad hoc usually requiring iterative LP and non-LP problem solving methods. We present the theory and methodology of quantile-DEA (qDEA), similar in concept to quantile-regression, which enables the analyst to directly use LP to obtain efficiency metrics while specifying that no more than ψ-percent of data points can lie external to the efficiency hull. Estimated qDEA-α frontiers encompassing proportion α = 1 − ψ of the data observations are contrasted to order-α frontier estimates. Quantile DEA is shown to be useful in addressing outliers in a study examining changes in relative state level agricultural efficiency measures over time.
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Notes
- 1.
The terminology qDEA denotes quantile DEA not to be confused with Kousmanen and Post (2002) Quadratic DEA which they denote QDEA.
- 2.
We wish to emphasize that while the qDEA process utilizes the results from a partial moment stochastic inequality to endogenously identify a set of external or “superefficient” DMUs, the set of external points are not randomly selected. This issue is discussed in more depth later in the paper.
- 3.
The resulting portfolios were found in many cases to be conservative due to the use of the LPM inequality but the resulting solutions did satisfy the requirement that no more than ψN of the income constraints were violated. Subsequent unpublished research suggested that a two stage process similar to the two-stage qDEA process discussed below would generate less conservative outcomes while still satisfying the desired limit on the number of income observations falling below g.
- 4.
- 5.
Setting ψ < 1/N will guaranty that no points will lie outside the hull i.e., obtain the conventional DEA results.
- 6.
We use the terminology “support points” to denote the points that define the hyperplane onto which the given DMU’s input-output points are projected with the distance from the initial point to the hyperplane being the estimated efficiency score. By a given DMU’s “reference set” we refer to the set of points remaining on the same side of the projection hyperplane as the given DMU.
- 7.
Table 1 lists the input and out values used in this example.
Table 1 CST input orientation example – input-output combinations, CRS and VRS DEA and qDEA efficiency scores for one and two external points - 8.
The fact that the green point is projected using each of the green circled points is readily derived by examining the dual solution to system (5) where the resulting dual or z j values are projection weights. The dual values for all constraining equations in system (5) will be non-zero valued. The constraints associated with all external points as well the new qDEA support points will be binding in system (5).
- 9.
R code available from the authors can be used to obtain qDEA solutions with input, output, or the more general ddea DEA models.
- 10.
The authors, to date, have not been able to derive closed form expressions for the asymptotic properties of the qDEA estimates. Numerical procedures described by Geyer (2013), suggest that qDEA estimates appear to have many of the desirable features of the FDH related order-m and order-α estimators including root-n convergence and asymptotic normality. The authors have experimented extensively with the use of nCm bootstrapping as discussed by Politis et al. (1999, 2001), Geyer (2013) and Simar and Wilson (2011b) and have achieved nominal qDEA confidence interval coverage levels that compare favorably with conventional DEA results reported by Simar and Wilson.
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Atwood, J.A., Shaik, S. (2018). Quantile DEA: Estimating qDEA-alpha Efficiency Estimates with Conventional Linear Programming. In: Greene, W., Khalaf, L., Makdissi, P., Sickles, R., Veall, M., Voia, MC. (eds) Productivity and Inequality. NAPW 2016. Springer Proceedings in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-68678-3_14
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