A generalized multiplicative directional distance function for efficiency measurement in DEA

For measuring technical efficiency relative to a log-linear technology, a generalized multiplicative directional distance function (GMDDF) is developed using the framework of multiplicative directional distance function (MDDF). Furthermore, a computational procedure is suggested for its estimation. The GMDDF serves as a comprehensive measure of efficiency in revealing Pareto-efficient targets as it accounts for all possible input and output slacks. This measure satisfies several desirable properties of an ideal efficiency measure such as strong monotonicity, unit invariance, translation invariance, and positive affine transformation invariance. This measure can be easily implemented in any standard DEA software and provides the decision makers with the option of specifying preferable direction vectors for incorporating their decision-making preferences. Finally, to demonstrate the ready applicability of our proposed measure, an illustrative empirical analysis is conducted based on real-life data set of 20 hardware computer companies in India.     

Negative features of hyperbolic and directional distance models for technologies with undesirable outputs

In data envelopment analysis for environmental performance measurement the undesirable outputs are taken into account. Ones of the standard approaches for dealing with the undesirable outputs are the hyperbolic and the directional distance measures. They both allow a simultaneous expansion of desirable outputs and a contraction of undesirable outputs by means of a single parameter. To meet environmental requirements, a technology with no disposability of undesirable outputs is often considered and the outputs are assumed to be only weakly disposable. We show that the combination of this type of technology with the hyperbolic measure, (or with its linearization, which is a special type of the directional distance model) may lead to a misleading efficiency score of the unit under evaluation. We derive the dual of the hyperbolic model under the environmental technology and describe some of its properties. Then, we use the hyperbolic and directional distance dual models for developing a second-phase method. This enables to detect the misleading scores of the decision making units. We illustrate the results on a real world data set.     

Cross-efficiency evaluation with directional distance functions

This paper extends the cross-efficiency evaluation for use with directional distance functions. Cross-efficiency evaluation has been developed with oriented Data Envelopment Analysis (DEA) models, so the extension proposed here is aimed at providing a peer-evaluation of decision making units (DMUs) based on measures that account for the inefficiency both in inputs and in outputs simultaneously. We explore the duality relations regarding the models of directional distance functions and define the cross-efficiencies on the basis of the equivalences with some fractional programming problems. Finally, we address in this new context the problem with the alternate optima for the weights and propose some models that implement different alternative secondary goals.     

Assessing eco-efficiency with directional distance functions

Eco-efficiency is a matter of concern at present that is receiving increasing attention in political, academic and business circles. Broadly speaking, this concept refers to the ability to create more goods and services with less impact on the environment and less consumption of natural resources, thus involving both economic and also ecological issues. In this paper we propose the use of directional distance functions and Data Envelopment Analysis techniques to assess eco-efficiency. More specifically, we show how these functions can be used to compute a wide range of indicators representing different objectives regarding economic and ecological performance. This methodological approach is applied to a sample of Spanish olive-growing farms to illustrate its great potential to provide policymakers and farm managers with sound information as a basis for strategic decision making. We also suggest further avenues to explore in this burgeoning line of research.     

The directional distance function and measurement of super-efficiency: an application to airlines data

In a recent paper published in this Journal, Lovell and Rouse (LR) proposed a modification of the standard data envelopment analysis (DEA) model that overcomes the infeasibility problem often encountered in computing super-efficiency. In the LR procedure one appropriately scales up the observed input vector (scale down the output vector) of the relevant super-efficient firm thereby usually creating its inefficient surrogate. By contrast, Chen suggested a different procedure that replaces input–output bundles that are found to be inefficient in standard DEA by their efficient projections. An alternative procedure proposed in this paper uses the directional distance function and the resulting Nerlove–Luenberger measure of super-efficiency. The fact that the directional distance function combines, by definition, features of both an input-oriented and an output-oriented model, generally leads to a complete ranking of the observations and is easily interpreted. A dataset on international airlines is utilized in an illustrative empirical application.     

Cost,revenue and profit efficiency measurement in DEA: A directional distance function approach

Estimation of efficiency of firms in a non-competitive market characterized by heterogeneous inputs and outputs along with their varying prices is questionable when factor-based technology sets are used in data envelopment analysis (DEA). In this scenario, a value-based technology becomes an appropriate reference technology against which efficiency can be assessed. In this contribution, the value-based models of Tone (2002) are extended in a directional DEA set up to develop new directional cost- and revenue-based measures of efficiency, which are then decomposed into their respective directional value-based technical and allocative efficiencies. These new directional value-based measures are more general, and include the existing value-based measures as special cases. These measures satisfy several desirable properties of an ideal efficiency measure. These new measures are advantageous over the existing ones in terms of (1) their ability to satisfy the most important property of translation invariance; (2) choices over the use of suitable direction vectors in handling negative data; and (3) flexibility in providing the decision makers with the option of specifying preferable direction vectors to incorporate their preferences. Finally, under the condition of no prior unit price information, a directional value-based measure of profit inefficiency is developed for firms whose underlying objectives are profit maximization. For an illustrative empirical application, our new measures are applied to a real-life data set of 50 US banks to draw inferences about the production correspondence of banking industry.     

A conditional directional distance function approach for measuring regional environmental efficiency: Evidence from UK regions

This paper, by using conditional directional distance functions as introduced by Simar and Vanhems [J. Econometrics 166 (2012) 342–354] modifies the model by Färe and Grosskopf [Eur. J. Operat. Res. 157 (2004) 242–245] and examines the link between regional environmental efficiency and economic growth. The proposed model using conditional directional distance functions incorporates the effect of regional economic growth on regions’ environmental efficiency levels. The results from UK regional data reveal a negative relationship between regions’ GDP per capita and environmental inefficiency up to a certain GDP per capita level. After that level it appears that the relationship becomes positive. As an overall result the regional environmental inefficiency-GDP per capita relationship appears to have a ‘U’ shape form.     

A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation

In this paper we focus on scale elasticity measure based on directional distance function for multi-output–multi-input technologies, explore its fundamental properties and show its equivalence with the input oriented and output oriented scale elasticity measures. We also establish duality relationship between the scale elasticity measure based on the directional distance function with scale elasticity measure based on the profit function. Finally, we discuss the estimation issues of the scale elasticity based on the directional distance function via the DEA estimator.     

Scale directional distance function and its application to the measurement of eco-efficiency in the manufacturing sector

Directional distance function (DDF) is a recognized technique for measuring efficiency while incorporating undesirable outputs. This approach allows for desirable outputs to be expanded while undesirable outputs are contracted simultaneously. A drawback of the DDF approach is that the direction vector to the production boundary is fixed arbitrarily, which may not provide the best efficiency measure. Therefore, this study extends the previous framework of efficiency analysis to introduce a new slacks-based measure of efficiency called the scale directional distance function (SDDF) approach. This new approach determines the optimal direction to the frontier for each unit of analysis and provides dissimilar expansion and contraction factors to achieve a more reasonable eco-efficiency score. This new approach is employed to measure the eco-efficiency of the Malaysian manufacturing sector. In addition, the paper demonstrates the use of the new approach to establish target values for the reduction/expansion of outputs in order for the inefficient DMUs to achieve full eco-efficiency. The results indicate that Melaka, Pulau Pinang, Negeri Sembilan, Sabah, Sarawak and Labuan have attained full eco-efficiency while Terengganu is the least eco-efficient. The overall eco-efficiency of the manufacturing sector in Malaysia is 80.5 % with wide variations across the states.     

Bounded distance preserving surjections in the projective geometry of matrices

We examine surjective maps which preserve a fixed bounded distance in both directions on some classical dual polar spaces.     

Super-efficiency measurement under variable return to scale: an approach based on a new directional distance function

A modified super-efficiency model based on the directional distance function (DDF) has recently been developed in order to tackle the infeasibility issue in the two exceptions of the Nerlove–Luenberger (N–L) super-efficiency model under variable return to scale (VRS). However, we find that model does not fully eliminate the infeasibility issue. This paper chooses an appropriate reference bundle in the DDF so that the resulting DDF-based VRS super-efficiency model is always feasible. The proposed new model successfully addresses the infeasibility issue of conventional VRS super-efficiency models and fully eliminates the infeasibility issue in the two exceptions of the VRS N–L super-efficiency model. Additional advantages of the new model include: it is unit-invariant and does not need to predetermine any parameter. Theoretical analyses and numerical examples support the practicality and superiority of our model when compared with other super-efficiency models.     

Estimating the technical efficiency of health care systems: A cross-country comparison using the directional distance function

Economic activity produces not only desirable outputs but also undesirable outputs. Undesirable outputs are usually omitted from efficiency assessments (i.e., applications of Data Envelopment Analysis) which fail to express the true production process. The directional distance function model has been used for handling asymmetrically both desirable and undesirable outputs in the assessment process. In the present paper, we apply a generalized directional distance function to measure the efficiency of the health systems of 171 countries. We incorporate both desirable and undesirable outputs into the efficiency assessment without transforming the latter type of outputs into inputs or into their inverse form, as is done in most of the extant studies that deal with the measurement of health efficiency. The methodology that we apply introduces a modified definition of the efficiency score which yields results consistent with those obtained from radial DEA models. In addition, our results are independent of the length of the direction vector.     

Minimum distance method for directional data and outlier detection

In this paper, we propose estimators based on the minimum distance for the unknown parameters of a parametric density on the unit sphere. We show that these estimators are consistent and asymptotically normally distributed. Also, we apply our proposal to develop a method that allows us to detect potential atypical values. The behavior under small samples of the proposed estimators is studied using Monte Carlo simulations. Two applications of our procedure are illustrated with real data sets.     

A directional distance based super-efficiency DEA model handling negative data

This paper develops a new radial super-efficiency data envelopment analysis (DEA) model, which allows input–output variables to take both negative and positive values. Compared with existing DEA models capable of dealing with negative data, the proposed model can rank the efficient DMUs and is feasible no matter whether the input–output data are non-negative or not. It successfully addresses the infeasibility issue of both the conventional radial super-efficiency DEA model and the Nerlove–Luenberger super-efficiency DEA model under the assumption of variable returns to scale. Moreover, it can project each DMU onto the super-efficiency frontier along a suitable direction and never leads to worse target inputs or outputs than the original ones for inefficient DMUs. Additional advantages of the proposed model include monotonicity, units invariance and output translation invariance. Two numerical examples demonstrate the practicality and superiority of the new model.     

Energy and CO2 emission performance in electricity generation: A non-radial directional distance function approach

This paper presents a non-radial directional distance function approach to modeling energy and CO₂emission performance in electricity generation from the production efficiency point of view. We first define and construct the environmental production technologies for the countries with and without CHP plants, respectively. The non-radial direction distance function approach is then proposed and several indexes are developed to measure energy and CO₂ emission performance of electricity generation. The directional distance functions established can be computed by solving a series of data envelopment analysis models. We then conduct an empirical study using the dataset for over one hundred countries. It is found that OECD countries have better carbon emission performance and integrated energy-carbon performance than non-OECD countries in electricity generation, while the difference in energy performance is not significant.     

Hellinger distance estimation of SSAR models

The present paper deals with the statistical inference of the simultaneous switching autoregressive (SSAR) model. This model has been introduced by Kunitomo and Sato (Jpn. Econ. Rev. 50 (2) (1996) 161) in order to take into account the asymmetry in financial and economical time series modelling. Under some conditions which ensure some probabilistic properties of the model, we establish, under other mild assumptions, the asymptotic properties of the minimum Hellinger distance estimates of the parameters. An application to a true data is also given.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.