The directional distance function and the translation invariance property
研究了在可变规模报酬下，方向距离函数的平移不变性条件，指出该性质依赖于方向向量的选择，并给出了方向向量需满足的充要条件。
Recently, in a Data Envelopment Analysis (DEA) framework, Färe and Grosskopf [15] argued that the input directional distance function is invariant to affine data transformations under variable returns to scale (VRS), which includes, as a particular case, the property of translation invariance. In this paper we show that, depending on the directional vector used, the translation invariance may fail. In order to identify the directional distance functions (DDFs) that are translation invariant under VRS, we establish a necessary and sufficient condition that the directional vector must fulfill. As a consequence, we identify the characteristics that the DDFs should verify to be translation invariant. We additionally show some distinguished members that satisfy the aforementioned condition. We finally give several examples of DDFs, including input and output DDFs, which are not translation invariant.