Chapter 3 Translation invariance in data envelopment analysis: A generalization

A slacks-based measure of efficiency in data envelopment analysis

“…Although this model can discriminate efficient and inefficient DMUs by the existence of slacks, it has no means to gauge the depth of inefficiency similar to B* in the CCR model. In an attempt to define inefficiency based on the slacks, Pastor (1996), Lovell and Pastor (1995), Cooper and Tone (1997), Thrall (1997) and others have proposed several formulae for finding a scalar measure. The following properties are considered as important in designing the measures.…”

A multilevel decomposition of school performance using robust nonparametric frontier techniques

“…There have been some advances in the field of nonparametric statistics that allow performance of statistical tests, in order to compare whether entire distributions show significant differences-differences that are not restricted to a few moments within the distributions. These tests were introduced by Li (1996) and by several applications, such as those in Murillo-Melchor et al (2010), Pastor andTortosa-Ausina (2008), andBalaguer-Coll et al (2010). 8 Results are displayed in Table 8, which reports the results of testing the null hypothesis that the distributions of each of the variables are the same for pairs of types of schools compared.…”

“…Since nonparametric frontier models cannot handle negative values, the latent variable scores were transformed so that we had only positive values (Pastor, 1996). The variable "school's average socioeconomic and cultural level" (x 3 ) corresponds to the arithmetic mean of the variable "socioeconomic and cultural level of the student's family.…”

Improving discrimination in data envelopment analysis: PCA–DEA or variable reduction

“…4 The additive linear program is particularly useful because this formulation corresponds to the Pareto-Koopmans (mixed) definition of technical efficiency. It also possesses a translation invariance property 5 under variable returns-to-scale (VRS) (Pastor, 1996) and data may be non-positive without the need for transformation. Another property that is generally considered crucial in performance analysis is units invariance.…”

“…Using principal components in place of original data does not affect the properties of the DEA models. For instance, the input-oriented, variable returnsto-scale, radial estimators continue to be units invariant and translation invariant with respect to outputs (Pastor (1996)). Principal components (PCs) represent the selection of a new coordinate system obtained by rotating the original system with x 1 ,…,x m as the coordinate axes.…”