Kristiaan Kerstens
email: k.kerstens@ieseg.fr

Jafar Sadeghi
email: j.sadeghi1987@gmail.com

Mehdi Toloo
email: mehdi.toloo@vsb.cz

Ignace Van De Woestyne
email: ignace.vandewoestyne@kuleuven.be

Procedures for ranking technical and cost efficient units: With a focus on nonconvexity

Keywords: Data Envelopment Analysis, Free Disposal Hull, Technical Efficiency, Cost Efficiency, Super-efficiency

This contribution extends the literature on super-efficiency by focusing on ranking cost-efficient observations. To the best of our knowledge, the focus has always been on technical super-efficiency and this focus on ranking cost-efficient observations may well open up a new topic. Furthermore, since the convexity axiom has both an impact on technical and cost efficiency, we pay a particular attention to the effect of nonconvexity on both super-efficiency notions. Apart from a numerical example, we use a secondary data set guaranteeing replication to illustrate these efficiency and super-efficiency concepts. Two empirical conclusions emerge. First, the cost super-efficiency notion ranks differently from the technical super-efficiency concept. Second, both cost and technical super-efficiency notions rank differently under convex and nonconvex technologies.

Introduction

The seminal articles of [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF] and [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF] have contributed to making the nonparametric approach to production theory one of the grand success stories in the economics and operations research (OR) literatures in terms of both methodological developments and empirical applications. An early bibliographical survey article counts about 800 published articles and dissertations related to the so-called Data Envelopment Analysis (DEA) literature over the years [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF]-1996(see Seiford (1997))). One of the most recent bibliographic reviews of [START_REF] Emrouznejad | A Survey and Analysis of the First 40 Years of Scholarly Literature in DEA: 1978-2016[END_REF] surveys the first 40 years of scholarly literature in DEA over the period 1978 till 2016 and lists about 10300 research articles. Empirical studies on efficiency and productivity using so-called frontier specifications are abundantly available and these frontier methodologies have become standard empirical tools that serve a variety of academic, regulatory and managerial purposes. Their widespread application in the academic literature analysing private and public sector performance-related issues can be glanced from, e.g., the [START_REF] Liu | Data Envelopment Analysis 1978-2010: A Citation-Based Literature Survey[END_REF] survey of empirical frontier applications. But, also the implementation of incentive regulatory mechanisms (e.g. price cap regulation) using frontier-based performance benchmarks is, for instance, rather widespread in countries having liberalized their network industries (e.g., for the electricity industry, one may consult the survey by [START_REF] Jamasb | Benchmarking and Regulation: International Electricity Experience[END_REF]). Finally, an example of a managerial application is the [START_REF] Sherman | Managing Bank Productivity Using Data Envelopment Analysis (DEA)[END_REF] study documenting how a US bank employs a basic frontier model to target sufficient savings in its branch network to fund its own expansion strategy.

While [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF] provided the first measurement scheme for the evaluation of technical and allocative efficiency resulting in cost efficiency in a frontier context, Färe, Grosskopf, and Lovell (1985, pp. 3-5) offer a more extended efficiency taxonomy by adding a scale efficiency component as well as a congestion component. In general, technical efficiency is solely based on information regarding physical inputs and outputs in the production process. The same holds true for congestion.

By contrast, to evaluate cost efficiency one also needs information regarding input prices in addition to physical inputs and outputs. Scale efficiency can be measured in a technical way based on inputs and outputs alone, but very early on also a cost-based approach was suggested that also requires information on input prices (see, e.g., [START_REF] Seitz | The Measurement of Efficiency Relative to a Frontier Production Function[END_REF]Seitz ( , 1971))). However, the variations on this basic measurement scheme are not central to our research questions.

The article of [START_REF] Andersen | A Procedure for Ranking Efficient Units in Data Envelopment Analysis[END_REF] is the first to ask a question with regard to the many observations that may obtain a similar relative technical efficiency status of unity. If all these observations are apparently equally technical efficient, is there any way to differentiate between these? [START_REF] Andersen | A Procedure for Ranking Efficient Units in Data Envelopment Analysis[END_REF]Petersen (1993, p. 1262) answer in the affirmative and state that "The basic idea is to compare the unit under evaluation with a linear combination of all other units in the sample, i.e., the DMU (Decision Making Unit) itself is excluded. It is conceivable that an efficient DMU may increase its input vector proportionally while preserving efficiency. The unit obtains in that case an efficiency score above one. The score reflects the radial distance from the DMU under evaluation 1 to the production frontier estimated with that DMU excluded from the sample, i.e., the maximum proportional increase in inputs preserving efficiency." This basic idea has led to the so-called superefficiency literature which investigates technical efficiency when the unit under evaluation is discarded from the technology. 1 This has led to explore issues of infeasibility under certain assumptions on technology and certain measurement orientations (see the surveys of Angulo-Meza and Estellita Lins (2002) and [START_REF] Adler | Ranking Methods Within Data Envelopment Analysis[END_REF]).

In addition to ranking efficient observations, super-efficiency models have also been used to develop tests for influential observations or outliers (see, e.g., [START_REF] Banker | The Super-Efficiency Procedure for Outlier Identification, Not for Ranking Efficient Units[END_REF], among many others).

This same idea of super-efficiency has also been explored within the context of incentive-based regulation theory employing frontier methodologies (see Bogetoft (2000) and [START_REF] Agrell | Regulatory Benchmarking: Models, Analyses and Applications[END_REF] for an early and a more recent survey). As far as we are aware of, Bogetoft (1994, p. 962) provides the first article within which a technical super-efficiency is defined to guarantee a proper incentive system. Agrell, Bogetoft, and Tind (2002, p. 6-7) similarly formulate a cost frontier from which the unit being evaluated is excluded from the technology definition. It seems that the existence of solutions for these technical and cost-based super-efficiency measures has not been given sufficient attention, especially in this incentive-based regulation theory employing frontier methodologies.

A first innovation of this contribution is to propose a super-efficiency model for the cost efficient observations. While cost efficiency is a sufficient condition for being technically efficient, technical efficiency is only a necessary but not a sufficient condition to being cost efficient. Thus, while there are fewer cost efficient than technically efficient observations, one could also wonder whether there is a way to discriminate between several of these cost efficient observations. This question seems never to have been treated in the literature.

A second innovation of this contribution is to focus on the impact of the convexity axiom on technical and cost super-efficiency models alike in the nonparametric, deterministic frontier tradition. 2 While it is tradition to impose convexity on the technology as a maintained axiom (see the seminal contributions of [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF] and [START_REF] Charnes | Measuring the Efficiency of Decision Making Units[END_REF]), [START_REF] Afriat | Efficiency Estimation of Production Functions[END_REF] is probably the first to mention a nonconvex Free Disposal Hull (FDH) model imposing only the assumptions of strong (free) disposal of inputs and outputs. This first nonconvex single output specification is generalized to the general multiple outputs case in the book chapter of [START_REF] Deprins | Measuring Labor Efficiency in Post Offices[END_REF]. [START_REF] Kerstens | Estimating Returns to Scale Using Nonparametric Deterministic Technologies: A New Method Based on Goodness-of-Fit[END_REF] extend this basic FDH model by introducing specific returns to scale assumptions and by proposing a new goodness-of-fit method to characterize returns 1 [START_REF] Banker | The Super-Efficiency Procedure for Outlier Identification, Not for Ranking Efficient Units[END_REF] mention an unpublished paper by Banker and Gifford from 1988 that delivered already the same idea.

2 In the stochastic parametric and semi-parametric frontier models, all firms are inefficient since inefficiency is nonnegative and the probability that inefficiency is exactly zero equals zero. Hence, there is no ranking problem of efficient observations. [START_REF] Kumbhakar | A Zero Inefficiency Stochastic Frontier Model[END_REF] is the seminal article allowing to represent a mixture of both fully efficient and inefficient firms. This basic idea has been extended in several directions. However, we are unaware of the development of a super-efficiency notion in this part of the frontier literature.

to scale for nonconvex technologies. [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF] define nonconvex cost functions corresponding to the specific returns to scale assumptions in [START_REF] Kerstens | Estimating Returns to Scale Using Nonparametric Deterministic Technologies: A New Method Based on Goodness-of-Fit[END_REF].

Apart from these basic contributions, one can mention a variety of recent contributions to the further development of similar nonconvex production models. For instance, [START_REF] Tavakoli | Free Disposal Hull Efficiency Scores of Units with Network Structures[END_REF] are the first to extend network models to the class of nonconvex technologies as defined in [START_REF] Kerstens | Estimating Returns to Scale Using Nonparametric Deterministic Technologies: A New Method Based on Goodness-of-Fit[END_REF]. [START_REF] Esteve | Efficiency Analysis Trees: A New Methodology for Estimating Production Frontiers Through Decision Trees[END_REF] combine regression trees from machine learning with the basic FDH model to estimate a new production frontier satisfying free disposability.

Most researchers are obviously aware of the fact that the convexity assumption affects the incidence and amount of technical inefficiency: under convexity less observations are technically efficient and the amount of technical inefficiency is higher. However, few people seem to realise that convexity also has consequences for the cost function: [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF] prove that convex cost functions are always smaller or equal to their convex counterparts for a given returns to scale assumption. There is only equality between convex and nonconvex cost functions in the case of a single output and constant returns to scale. In other words, the convexity assumption also affects the incidence and amount of cost inefficiency. Kerstens and Van de Woestyne (2021) empirically document the very substantial impact of convexity on cost function estimates in the literature (a total of 14 studies) as well as on two secondary data sets. In a similar vein, Kerstens, Sadeghi, and Van de Woestyne (2019) empirically document the substantial effect of convexity on technical efficiency and cost function based primal and cost-based capacity concepts. This fact that convexity affects the incidence and amounts of all sorts of inefficiency makes the issue relevant in a super-efficiency model context. Apart from the early contribution of [START_REF] Van Puyenbroeck | Some Remarks on Modified FDH[END_REF] and the more recent one by [START_REF] Aldamak | Dual Frontiers Without Convexity[END_REF] focusing on technical super-efficiency under nonconvexity, we are unaware of any work focusing on the impact of convexity on technical and cost super-efficiency from a comparative perspective. This contribution is structured as follows. Section 2 starts with the basic definitions of the technology, efficiency measures and the cost function. It also develops some basic efficiency decompositions.

Then, to show the versatility of the approach, we discuss cost efficiency with incomplete information as introduced in [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF][START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF]. Section 3 starts from the case of technical super-efficiency to introduce the new notion of cost super-efficiency and the ensuing super-efficiency decompositions. Thereafter, we also define cost super-efficiency for the incomplete information case.

Both the technical and cost super-efficiency definitions are numerically illustrated with a figure in Section 4. Key results on technical and cost super-efficiency are developed in Section 5 and partially illustrated with numerical illustrations. An empirical illustration based on a secondary data set is developed in Section 6. We finish with some conclusions in the final Section 7.

2 Technology and Cost Function: Basic Definitions and Decompositions

Technology and Cost Function: Basic Definitions

In this section, we define technology and some basic notation. Given N -dimensional input vectors

x ∈ R N + and M -dimensional output vectors y ∈ R M + , the production possibility set or technology T can be defined as T = {(x, y) | x can produce at least y}. It is common to impose some conditions on the input and output data defining the technology: (i) each producer uses nonnegative amounts of each input to produce nonnegative amounts of each output; (ii) there is an aggregate production of positive amounts of every output, and an aggregate use of positive amounts of every input; and (iii) each producer uses a positive amount of at least one input to produce a positive amount of at least one output (see, e.g., Färe, Grosskopf, and Lovell (1994, p. 44-45)). The input set L(y) = {x | (x, y) ∈ T } associated with T holds all input vectors x capable of producing at least a given output vector y. In a similar way, the output set P (x) = {y | (x, y) ∈ T } associated with T holds all output vectors y that can be produced from at most a given input vector x.

Throughout this contribution, technology T satisfies some combination of the following standard assumptions:

(T.1) Possibility of inaction and no free lunch, i.e., (0, 0) ∈ T and if (0, y) ∈ T , then y = 0.

(T.2) T is a closed subset of R N + × R M + .
(T.3) Strong input and output disposal, i.e., if (x, y) ∈ T and (x , y

) ∈ R N + × R M + , then (x , -y ) ≥ (x, -y) ⇒ (x , y ) ∈ T . (T.4) (x, y) ∈ T ⇒ δ(x, y) ∈ T for δ ∈ Γ, where: (i) Γ ≡ CRS = {δ | δ ≥ 0}; (ii) Γ ≡ VRS = {δ | δ = 1}. (T.5) T is convex.
Commenting these classical assumptions on the production technology, one can recall the following (see, e.g., [START_REF] Hackman | Production Economics: Integrating the Microeconomic and Engineering Perspectives[END_REF] for details). Inaction is possible and there exists no free lunch.

Technology is a closed set. We assume free or strong disposability of both inputs and outputs in that inputs can be destroyed and outputs can be discarded at no opportunity costs. We allow independently for two returns to scale assumptions: either constant returns to scale (CRS), of variable returns to scale (VRS). Finally, the technology set is convex. Not all these axioms are maintained in the empirical analysis. 3 In particular, key assumptions distinguishing some of the technologies in the empirical analysis are CRS versus VRS, and convexity versus nonconvexity.

The input distance function completely characterizes the input set L(y) and is defined as follows:

D i (x, y | T ) = max λ {λ | λ ≥ 0, (x/λ, y) ∈ T } = max λ {λ | λ ≥ 0, x/λ ∈ L(y)}.
(1)

Its main properties are: (i) D i (x, y | T ) ≥ 1, with efficient production on the boundary (isoquant) of L(y) represented by unity; (ii) it has a cost interpretation (see, e.g., [START_REF] Hackman | Production Economics: Integrating the Microeconomic and Engineering Perspectives[END_REF]). The inverse of this input distance function

DF i (x, y | T ) = [D i (x, y | T )]
-1 is known as the radial input efficiency measure. Therefore, this radial input efficiency measure is defined as:

DF i (x, y | T ) = min λ {λ | λ ≥ 0, (λx, y) ∈ T } = min λ {λ | λ ≥ 0, λx ∈ L(y)}. (2)
Obvisously, it is situated between zero and unity (0 < DF i (x, y) ≤ 1), with efficient production on the boundary (isoquant) of the input set L(y) represented by unity.

Looking to a dual representation of technology, the cost function is defined as the minimum expenses required to produce a given output vector y for a given vector of semi-positive input prices

(w ∈ R N + ): C(y, w | T ) = min x {wx | (x, y) ∈ T } = min x {wx | x ∈ L(y)}. (3)
Duality relations link these primal and dual representations of technology. Duality allows a wellbehaved technology to be reconstructed from the observations on cost minimizing producer behavior, and the reverse. The duality between input distance function (1) and cost function (3) can be stated as follows:

D i (x, y | T ) = min w {wx | C(y, w | T ) ≥ 1}, x ∈ L(y), (4)
C(y, w | T ) = min x {wx | D i (x, y | T ) ≥ 1}, w > 0. ( 5
)
It is common to establish such duality relations under the hypothesis of a convex technology or a convex input set (e.g., (Hackman, 2008, Ch. 7)). [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF] are the first to establish a local duality result between nonconvex technologies subject to various scaling laws and their corresponding nonconvex cost functions.

Nonparametric Frontier Technology and Cost Function Specification

For the methodological results and the empirical application, we assume a convex or nonconvex, nonparametric frontier technology under VRS and CRS assumptions. Let K input-output combinations

(x k , y k ) ∈ R N + × R M + (k ∈ {1, .
. . , K}) define the technology, then a unified algebraic representation of the corresponding convex and nonconvex nonparametric frontier technologies under the VRS and CRS assumptions is as follows:

T Λ,Γ = (x, y) | x ≥ K k=1 x k δz k , y ≤ K k=1 y k δz k , z = (z 1 , . . . , z K ) ∈ Λ, δ ∈ Γ , (6)
where

(i) Γ ≡ CRS = {δ | δ ≥ 0}; (ii) Γ ≡ VRS = {δ | δ = 1}; and (i) Λ ≡ C = z | K k=1 z k = 1 and ∀k ∈ {1, . . . , K} : z k ≥ 0 ; (ii) Λ ≡ NC = z | K k=1 z k = 1 and ∀k ∈ {1, . . . , K} : z k ∈ {0, 1} .
The activity vector z operates subject to a convexity (Λ ≡ C) or a nonconvexity (Λ ≡ NC) constraint.

The activity vector z of real numbers summing to unity represents the convexity axiom, while this same sum constraint with each vector element being a binary represents nonconvexity. There is also a scaling parameter (δ) allowing for a particular scaling of all K observations determining the technology: this scaling parameter is free under CRS and fixed at unity under VRS. [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF] demonstrate that nonconvex CRS and VRS technologies satisfy a minimum extrapolation principle.

To compute the input efficiency measure (2) relative to convex technologies in (6) requires solving a nonlinear programming (NLP) problem for each evaluated observation. However, Briec and Kerstens (2006, Lemma 2.1) show how this NLP can be transformed to a linear programming (LP)

problem by substituting zk = δz k in (6). For the nonconvex technologies, nonlinear binary mixed integer programs must be solved, but alternative solution strategies (including implicit enumeration) are available (see Kerstens and Van de Woestyne ( 2014)).

Assuming the nonparametric frontier technology T Λ,Γ , the cost function (3) defined as the minimum expenditures needed to produce a given output vector y p ∈ R M + for a given vector of semipositive common input prices w ∈ R N + is obtained by solving the following programming problem [START_REF] Färe | The Measurement of Efficiency of Production[END_REF]):

C(y p , w | T Λ,Γ ) = min x {wx | (x, y p ) ∈ T Λ,Γ } = min x {wx | x ∈ L(y p )}. ( 7
)
Computing the cost function relative to convex technologies is straightforward when transforming the NLP to the corresponding LP problem as indicated above. Computing the cost function relative to nonconvex technologies can be done using implicit enumeration algorithms specified in [START_REF] Briec | Non-convex Technologies and Cost Functions: Definitions, Duality and Nonparametric Tests of Convexity[END_REF].

Overall or Cost Efficiency Decomposition

Following [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF], the following basic overall or cost efficiency measure lends itself to a decomposition into technical and allocative efficiencies. A natural measure for cost efficiency is to take the ratio of minimum to actual observed cost:

CE(x p , y p , w | T Λ,Γ ) = C(y p , w | T Λ,Γ ) wx p . ( 8
)
Similar to technical efficiency, cost efficiency is situated between zero and unity: i.e., 0

< CE (x p , y p , w | T Λ,Γ ) ≤ 1.
Allocative efficiency is then defined as a ratio of cost efficiency over technical efficiency. Hence, the allocative efficiency of observation (x p , y p ) is defined as a residual as follows:

AE(x p , y p , w | T Λ,Γ ) = CE(x p , y p , w | T Λ,Γ ) DF i (x p , y p | T Λ,Γ ) . (9)
Note that the above components are defined for convex and nonconvex technologies under both CRS and VRS and it is easy to see that the allocative efficiency component always exceeds the cost efficiency measure: i.e., AE(x p , y p , w | T Λ,Γ ) ≥ CE(x p , y p , w | T Λ,Γ ).

While [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF] already explores the notion of scale efficiency, [START_REF] Färe | The Measurement of Efficiency of Production[END_REF] define scale efficiency by a ratio of technical efficiency under CRS over technical efficiency under VRS. Hence the scale efficiency of observation (x p , y p ) is defined as follows:

SCE(x p , y p | T Λ,. ) = DF i (x p , y p | T Λ,CRS ) DF i (x p , y p | T Λ,V RS ) . (10)
Clearly, scale efficiency is situated between zero and unity, i.e., 0 < SCE(x p , y p | T Λ,. ) ≤ 1. The superscript Λ indicates that scale efficiency can be defined under both convex and nonconvex cases.

Alternatively, Seitz (1971) defines a cost-based scale efficiency component as the ratio of cost functions under CRS and VRS.

Based on ( 9) and ( 10), CRS technical efficiency can be decomposed into VRS technical efficiency and scale efficiency: 11) [START_REF] Färe | The Measurement of Efficiency of Production[END_REF] refine this decomposition by also distinguishing between pure technical efficiency and structural efficiency or congestion efficiency. Other decompositions are available in the literature: examples include [START_REF] Aparicio | How to Properly Decompose Economic Efficiency using Technical and Allocative Criteria with Non-Homothetic DEA Technologies[END_REF] and [START_REF] Aparicio | Measuring and Decomposing Profit Inefficiency Through the Slacks-Based Measure[END_REF].

CE(x p , y p , w | T Λ,CRS ) = DF i (x p , y p | T Λ,CRS ) × AE(x p , y p , w | T Λ,CRS ) = DF i (x p , y p | T Λ,V RS ) × SCE(x p , y p | T Λ,. ) × AE(x p , y p , w | T Λ,CRS ). (

Overall or Cost Efficiency with Incomplete Prices

When price information is incomplete, [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF][START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF] derive upper and lower bounds for cost efficiency and allocative efficiency assuming incomplete price data in the form of a convex polyhedral cone. For this purpose, [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF] use an equivalent formulation of cost efficiency (8) which represents cost efficiency at normalised prices:

CE(x p , y p , w | T Λ,Γ ) = min x ∈L(y) {w x | w = αw, α ∈ R + , w x p = 1}. (12)
In ( 12) input prices are normalized such that the normalized cost of DMU p equals unity. To relax the information requirement on prices, [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF] present a convex polyhedral cone W ⊂ R N + as the price domain of inputs as follows:

W = {w ∈ R N + | Aw ≥ 0}, ( 13
)
where A is a L × N matrix composed of L row vectors A 1 , . . . , A L . Therefore, this cone W represents the input price domain by means of L linear inequalities.

To obtain an upper bound for cost efficiency, [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF] use the maximum value of cost efficiency over the price domain as follows:

CE(x p , y p | T Λ,Γ ) = max w∈W ( min x ∈L(yp)
{wx | wx p = 1}). ( 14) Kuosmanen and Post (2001, p. 52) prove that model ( 14) under input convexity, nonconvexity of outputs, and VRS can be equivalently obtained by solving the following model:

CE(x p , y p ) = min θ,z k ,β θ s.t k∈χ(yp) z k x k + βA ≤ θx p , k∈χ(yp) z k = 1, θ ≥ 0, β ∈ R L + , z k ≥ 0, k ∈ χ(y p ), (15)
where χ(y p ) = {k | y k ≥ y p } indicates all the observations that have a higher output level than y p .

Note that p ∈ χ(y p ), therefore χ(y p ) = ∅.

To obtain a lower bound for cost efficiency, [START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF][START_REF] Kuosmanen | Measuring Economic Efficiency with Incomplete Price Information: With an Application to European Commercial European[END_REF] use the minimum value of cost efficiency over the price domain as follows:

CE(x p , y p , w | T Λ,Γ ) = min w∈W ( min x ∈L(yp) {wx | wx p = 1}). ( 16
)
To solve model ( 16), Kuosmanen and Post (2003, p. 456) reverse the order of the two minimization problems as follows:

CE(x p , y p , w | T Λ,Γ ) = min x ∈L(yp) ( min w∈W {wx | wx p = 1}). ( 17
)
Therefore, we first need to solve model min w∈W {wx k | wx p = 1} for each k ∈ χ(y p ) and then we select the minimum of their optimal values as the lower bound of cost efficiency of DMU p (see section 4

in Kuosmanen and Post (2003, p. 456) for details).

Obviously, corresponding to these upper and lower bounds on overall or cost efficiency, it is straightforward to define a corresponding allocative efficiency component.

3 Super-Efficiency: Existing and New Definitions and Decompositions 3.1 Technical Super-Efficiency: Existing Definition [START_REF] Andersen | A Procedure for Ranking Efficient Units in Data Envelopment Analysis[END_REF] develop a procedure to rank technically efficient units. Their method enables an efficient observation (x p , y p ) to achieve an efficiency score greater than or equal to unity by removing this observation from those defining technology T Λ,Γ . Referring with subscript p to the exclusion of this p-th observation, the resulting technology can mathematically be formulated as follows:

T Λ,Γ p = (x, y) | x ≥ K k=1 x k δz k , y ≤ K k=1 y k δz k , z p = 0, z ∈ Λ, δ ∈ Γ . ( 18
)
Observe that the removal of the p-th observation is realized by forcing the p-th component of the activity vector to be zero. Consequently, this p-th observation no longer influences the constraints.

The traditional technical efficiency measure (2) can now be transformed in a super-efficiency context as follows:

Definition 3.1. The input-oriented technical super-efficiency model is defined as:

DF SE i (x p , y p | T Λ,Γ p ) = min λ {λ | λ ≥ 0, (λx p , y p ) ∈ T Λ,Γ p }, (19)
with T Λ,Γ p defined in (18). [START_REF] Andersen | A Procedure for Ranking Efficient Units in Data Envelopment Analysis[END_REF]Petersen, 1993, p. 1263)

Note that DF

i (x p , y p | T Λ,Γ ) = DF SE i (x p , y p | T Λ,Γ p ) for technically inefficient observations, while for technically efficient observations, DF i (x p , y p | T Λ,Γ ) ≤ DF SE i (x p , y p | T Λ,Γ p ) (see
| T Λ,Γ p ) > 1.
Note that the super-efficiency model DF SE i (x p , y p | T Λ,Γ p ) can lead to infeasibilities in practical applications. In fact, this observation has led to a rather substantial literature: the reader is referred to the surveys in Angulo-Meza and Estellita Lins ( 2002), [START_REF] Soltanifar | Survey on Rank Preservation and Rank Reversal in Data Envelopment Analysis[END_REF] and [START_REF] Adler | Ranking Methods Within Data Envelopment Analysis[END_REF] for further details.

Cost Super-Efficiency: New Definition

The previous section focused on the technical super-efficiency where input price information is not available. The super-efficiency concept seems never to have been applied in a cost function context. This section turns to the topic of cost efficiency to show how nonparametric frontiers can be used to identify a cost super-efficiency concept when information on input prices and costs are known exactly.

A natural measure for cost super-efficiency of observation (x p , y p ) is to take the ratio of the minimum cost when this observation is removed from the set of observations, relative to the actual observed cost. Therefore, we define the cost super-efficiency of (x p , y p ) by considering the production possibility set T Λ,Γ p defined in (18) as follows:

Definition 3.2. The cost super-efficiency of observation (x p , y p ) is defined by

CE SE (x p , y p , w | T Λ,Γ p ) = C SE (y p , w | T Λ,Γ p ) wx p , (20)
where C SE (y p , w | T Λ,Γ p ) is obtained as follows:

C SE (y p , w | T Λ,Γ p ) = min x {wx | x ≥ 0, (x, y p ) ∈ T Λ,Γ p }. ( 21
) Note that CE SE (x p , y p , w | T Λ,Γ p ) = CE(x p , y p , w | T Λ,Γ ) for cost inefficient observations, while CE(x p , y p , w | T Λ,Γ ) ≤ CE SE (x p , y p , w | T Λ,Γ
p ) for cost efficient observations. Consequently, observation (x p , y p ) is cost inefficient by definition (8) if and only if the cost super-efficiency estimate

CE SE (x p , y p , w | T Λ,Γ p ) < 1,

Overall or Cost Super-Efficiency Decomposition: New Definition

By analogy to Section 2.3, we now decompose the cost super-efficiency (20) into several efficiency components.

Allocative super-efficiency is defined as the ratio of cost super-efficiency over the technical superefficiency. Hence, the allocative super-efficiency of observation (x p , y p ) is defined as follows:

AE SE (x p , y p , w | T Λ,Γ p ) = CE SE (w p , y p | T Λ,Γ p ) DF SE i (x p , y p | T Λ,Γ p ) . ( 22
)
Note again that the above components are defined for convex and nonconvex technologies alike under both CRS and VRS.

Based on the CRS and VRS technical super-efficiency scores, we can define a new scale superefficiency by taking ratio of technical super-efficiency under CRS over technical super-efficiency under VRS. Hence, this scale super-efficiency of observation (x p , y p ) is defined as follows:

SCE SE (x p , y p | T Λ,. p ) = DF SE i (x p , y p | T Λ,CRS p ) DF SE i (x p , y p | T Λ,V RS p ) . (23)
Clearly, 0 < SCE SE (x p , y p | T Λ,. p ) ≤ 1. Similar to Seitz (1971), one could equally define a costbased scale super-efficiency component as the ratio of cost functions under CRS and VRS computed relative to technology (18). Furthermore, based on ( 22) and ( 23), CRS technical super-efficiency can be decomposed into VRS technical super-efficiency and scale super-efficiency as follows:

CE SE (w p , y p | T Λ,CRS p ) = DF SE i (x p , y p | T Λ,CRS p ) × AE SE (x p , y p , w | T Λ,CRS p ) = DF SE i (x p , y p | T Λ,V RS p ) × SCE SE (x p , y p | T Λ,. p ) × AE SE (x p , y p , w | T Λ,CRS p ). ( 24
)
Note that if observation (x p , y p ) is technically inefficient under VRS (i.e., DF i (x p , y p | T Λ,V RS ) < 1),

then DF i (x p , y p | T Λ,V RS ) = DF SE i (x p , y p | T Λ,V RS p ), DF i (x p , y p | T Λ,CRS ) = DF SE i (x p , y p | T Λ,CRS p ) and CE(w p , y p | T Λ,CRS ) = CE SE (w p , y p | T Λ,CRS p
). Hence, in this case, technical efficiency with technical super-efficiency, allocative efficiency with allocative super-efficiency and scale efficiency with scale super-efficiency are identical. Consequently, if the observation is technically inefficient under VRS, then its decomposition of overall or cost efficiency (11) and its decomposition of overall or cost super-efficiency (24) lead to the same results.

Overall or Cost Super-Efficiency with Incomplete Prices: New Definition

To calculate an upper bound of the cost super-efficiency under the maintained assumptions with respect to price domain W , we introduce the following model:

CE SE (x p , y p ) = min θ,z k ,β θ s.t k∈χ(xp) z k y k + βA ≤ θx p , k∈χ (yp)
z k = 1, θ ≥ 0, β ∈ R L + , z p = 0, z k ≥ 0, k ∈ χ(y p ), k = p, (25)
where χ(y p ) is defined as above. Observe that the removal of the p-th observation is implemented by forcing the p-th component of the activity vector to be equal to zero. Consequently, this p-th observation no longer influences the constraints. If we have χ(y p ) = {p}, then model ( 25) is infeasible and there is no upper bound for the cost super-efficiency of DMU p .

To measure the lower bound of the cost super-efficiency under the maintained assumptions with respect to price domain W , we introduce the following model:

CE SE (x p , y p ) = min
DF SE i (x B , y B | T C,V RS B ) = OB 2 OB > 1 and CE SE (x B , y B , w | T C,V RS B ) = OB 1 OB > 1, respectively. Since OB 2 > OB 1 , DF SE i (x B , y B | T C,V RS ) > CE SE (x B , y B , w | T C,V RS B
) > 1 meaning that observation B is technically and cost super-efficient. Now, we focus on observation D and assume it faces the same input prices as observation B.

For this observation D, the corresponding technical and cost efficiency measures are

DF i (x D , y D | T C,V RS ) = OD 1 OD < 1 and CE(x D , y D , w | T C,V RS ) = OD 2 OD < 1, respectively. Since OD 1 > OD 2 , CE(x D , y D , w | T C,V RS ) < DF i (x D , y D | T C,V RS
) < 1 from which we conclude that observation D is both technically and cost inefficient. In this case, the boundary of the input set L(y) for a given output level y, which is shown by the polyline ABC and its horizontal and vertical extensions at A and C respectively, remains unchanged after removing observation D from the observations defining technology. Thus, by removing this inefficient unit from technology, both technical and cost efficiency measures remain unchanged. Hence,

DF i (x D , y D | T C,V RS ) = DF SE i (x D , y D | T C,V RS D ) and CE(x D , y D , w | T C,V RS ) = CE SE (x D , y D , w | T C,V RS D
).

Technical and Cost Super-Efficiency: Key Results

A first Proposition 5.1 compares the technical and cost efficiency measures as well as the technical and cost super-efficiency measures:

Proposition 5.1. For every observation (x p , y p ) ∈ T Λ,Γ :

(i) CE(x p , y p , w | T Λ,Γ ) ≤ DF i (x p , y p | T Λ,Γ ) ≤ 1.
(ii) The number of cost efficient observations is smaller than or equal to the number of inputoriented technical efficient observations.

(iii) CE SE (x p , y p , w | T Λ,Γ p ) ≤ DF SE i (x p , y p | T Λ,Γ p ).
Proof. See Appendix A.

Proposition 5.1 shows that the cost efficiency and cost super-efficiency measures are always smaller than or equal to the technical efficiency and technical super-efficiency measures, respectively, in both convex and nonconvex cases and under both CRS and VRS. Moreover, properties (i) and (iii) of Proposition 5.1 directly lead to the following corollary:

Corollary 5.1.

(i) Allocative efficiency is situated between zero and unity, i.e., for every observation (x p , y p ) ∈ T Λ,Γ : 0 < AE(x p , y p , w | T Λ,Γ ) ≤ 1.

(ii) Allocative super-efficiency is situated between zero and unity, i.e., for every observation (x p , y p ) ∈

T Λ,Γ : 0 < AE SE (x p , y p , w | T Λ,Γ p ) ≤ 1.
The next Proposition 5.2 compares the technical efficiency and technical super-efficiency measures as well as the cost efficiency and cost super-efficiency measures in both convex and nonconvex cases and under both CRS and VRS:

Proposition 5.2. For every observation (x p , y p ):

(i) DF i (x p , y p | T C,Γ ) ≤ DF i (x p , y p | T N C,Γ ); (ii) DF SE i (x p , y p | T C,Γ p ) ≤ DF SE i (x p , y p | T N C,Γ p ); (iii) CE(x p , y p , w | T C,Γ ) ≤ CE(x p , y p , w | T N C,Γ ); (iv) CE SE (x p , y p , w | T C,Γ p ) ≤ CE SE (x p , y p , w | T N C,Γ p ).
Proof. See Appendix A.

When comparing convex and nonconvex results for all efficiency concepts (i.e., technical and cost efficiency as well as technical and cost super-efficiency measures), then the obtained results under the convexity assumption are always smaller than or equal to the ones obtained under the nonconvexity assumption. Note that Proposition 5.2(i) and (iii) is already proven in Briec, Kerstens, and Vanden Eeckaut (2004, p. 178). [START_REF] Zhu | Robustness of the Efficient DMUs in Data Envelopment Analysis[END_REF] indicates that the input-oriented super-efficiency model under CRS is feasible unless certain patterns of zero data entries are present in the inputs. Therefore, if one assumes that all inputs data are strictly positive, then the input-oriented super-efficiency CRS model is always feasible.

However, in Proposition 5.3(i), we prove that the cost super-efficiency model under CRS is always feasible even if there are some zero data entries in the inputs. Proposition 5.3 focuses on the potential for infeasibility for both technical and cost super-efficiency:

Proposition 5.3. (i) For every observation (x p , y p ) ∈ T Λ,CRS , the cost super-efficiency model ) is identical. Note that Proposition 5.3(ii) shows that the infeasibility of model CE SE (x p , y p , w | T Λ,V RS p ) under both convex and nonconvex cases are independent from the input prices. According to Proposition 5.3(iii) and (iv), if an observation results in an infeasibility under convexity, then this same observation also leads to an infeasibility under nonconvexity for both technical and cost super-efficiency measures. Note that the inverses of Proposition 5.3(iii) and (vi) are not satisfied. Consequently, the number of infeasible observations in the convex case is smaller than or equal to the number of infeasible observations in the nonconvex case for both technical and cost super-efficiency measures. The potential incidence of infeasibility is worse under nonconvexity because the volume of the technology is smaller than or equal to the volume of its convex counterpart. Proposition 5.3(v) shows that infeasibilities occur for the same observations in both upper and lower bounds models CE SE (x p , y p ) and CE SE (x p , y p ). Moreover, infeasibility of these two upper and lower bounds models are independent from the price domain of inputs W .

C SE (y p , w | T Λ,CRS p ) is always feasible. (ii) Model DF SE i (x p , y p | T Λ,V RS p ) is feasible if and only if model CE SE (x p , y p , w | T Λ,V RS p ) is feasible. (iii) If model DF SE i (x p , y p | T C,Γ p ) is infeasible, then model DF SE i (x p , y p | T N C,Γ p ) is infeasible. (iv) If model CE SE (x p , y p , w | T C,V RS p ) is infeasible, then model CE SE (x p , y p , w | T N C,V RS p ) is infeasible. (v)
While Agrell, Bogetoft, and Tind (2002, p. 6-7) define the cost frontier for CRS and VRS assumptions alike, it is clear that a super-efficiency cost frontier version may not exist under VRS, but it always exists under CRS. Unfortunately, this lack of feasibility under VRS is much more important from a policy viewpoint than this existence result under CRS, since few economists and regulators are advocating to impose CRS when determining a frontier-based cost norm. This existence result under CRS seems new to the incentive-based regulation theory employing frontier methodologies.

To illustrate the difference in the infeasibility behaviour between technical super-efficiency and However, cost minimization is still possible when observation E is removed and results in the cost-minimizing input vector located at D on the frontier of technology T C,CRS : this cost function is drawn in a dash dotted line. Hence, the corresponding cost super-efficiency measure of observation

E is simply CE SE (x E , y E , w | T C,CRS E ) = OE
OE > 1 and is perfectly feasible. and two outputs (upper horizontal plane). We focus on observation 4 indicated by a small sphere which is an inefficient unit under the convex case and an efficient unit under the nonconvex case. The visible production frontiers are those with observation 4 excluded: the nonconvex case is in a solid color, and the convex case is in a slightly transparent color. Observation 4 is scaled upward to the position indicated by the cube when optimizing in the input direction and assuming convexity. But, in the case of nonconvexity, there is no scaling up or down that reaches the boundary of technology with observation 4 excluded, whence the infeasibility.

DF i (x D , y D | T C,V RS ) = DF i (x D , y D | T N C,V RS ) = D 1 D D 1 D = 1 and CE(x D , y D , w | T C,V RS ) = CE(x D , y D , w | T N C,V RS ) = D 1 D D 1 D = 1, respectively. Therefore, we have CE(x D , y D , w | T Λ,V RS ) = DF i (x D , y D | T Λ,V RS ) = 1 for Λ = C and Λ = N C.
The problem of infeasibilities can lead to several reactions. One position that has been defended in the literature the infeasibility of the Luenberger productivity indicator is to accept the fact that distance functions need not always be defined and to simply report all infeasibilities whenever these occur (see, e.g., [START_REF] Briec | Infeasibilities and Directional Distance Functions with Application to the Determinateness of the Luenberger Productivity Indicator[END_REF]).The same argument could be applied to the cost function. This is the position taken in this contribution. Another position is to remedy the infeasibilities in distance function and/or cost function whenever this is possible. While some undeniable progress has been made to reduce or eliminate the infeasibility problem for distance functions (see supra), we are unaware of any solution to the infeasibility problem for the cost function. Moreover, depending on whether or not one attaches some importance to the static efficiency decomposition one may wish to have solutions that are structurally similar across different super-efficiency concepts. We think that it is desirable to find a solution to the infeasibility problem that works in a similar way for technical and cost super-efficiency concepts alike. It remains an open question whether this is possible.4 Proposition 5.4 compares the allocative and scale efficiency measures with the allocative and scale super-efficiency measures: Proposition 5.4. For every observation (x p , y p ):

(i) AE(x p , y p , w | T Λ,CRS ) ≥ AE SE (x p , y p , w | T Λ,CRS p ) (ii) SCE(x p , y p | T Λ,. ) ≥ SCE SE (x p , y p | T Λ,. p )
Proof. See Appendix A.

Proposition 5.4 shows that the allocative and scale efficiency measures are greater than or equal to the allocative and scale super-efficiency measures, respectively, under both convex and nonconvex cases.

Proposition 5.5 compares the allocative super-efficiency measure with the cost super-efficiency measure:

Proposition 5.5. For every observation (x p , y p ):

(i) If DF SE i (x p , y p | T Λ,V RS p ) ≤ 1, then AE SE (x p , y p , w | T Λ,CRS p ) ≥ CE SE (x p , y p , w | T Λ,CRS p ). (ii) If DF SE i (x p , y p | T Λ,V RS p ) > 1, then AE SE (x p , y p , w | T Λ,CRS p ) > = < CE SE (x p , y p , w | T Λ,CRS p ).
Proof. See Appendix A.

Note that the allocative efficiency measure is always higher than the cost efficiency measure, and Proposition 5.5 shows that we have the same result for allocative super-efficiency and cost superefficiency measures when

DF SE i (x p , y p | T Λ,V RS p ) ≤ 1. Also, if DM U p is a super-efficient unit, i.e., DF SE i (x p , y p | T Λ,V RS p
) > 1, then we can not compare the allocative super-efficiency and the cost super-efficiency measures.

Finally, Proposition 5.6 compares the scale super-efficiency measure with the cost super-efficiency measure:

Proposition 5.6. For every DM U p we have:

(i) If DF SE i (x p , y p | T Λ,V RS p ) ≤ 1, then SCE SE (x p , y p | T Λ,. p ) ≥ CE SE (x p , y p , w | T Λ,CRS p ). (ii) If DF SE i (x p , y p | T Λ,V RS p ) > 1, then SCE SE (x p , y p | T Λ,. p ) > = < CE SE (x p , y p , w | T Λ,CRS p ).
Proof. See Appendix A.

Recall that scale efficiency is always greater than or equal to cost efficiency. Proposition 5.6 proves that the same result holds true for scale super-efficiency and cost super-efficiency measures when

DF SE i (x p , y p | T Λ,V RS p ) ≤ 1. Also, if an observation is technically super-efficient (i.e., DF SE i (x p , y p | T Λ,V RS p ) > 1)
, then there is no relation whatsoever between the scale super-efficiency and the cost super-efficiency measures.

It is important to note that the results obtained in this contribution are not available in both the Van Puyenbroeck (1998) and [START_REF] Aldamak | Dual Frontiers Without Convexity[END_REF] articles. Furthermore, our results normally carry over when one would only consider partial convexity rather than nonconvexity. We now turn to an empirical illustration of most of the obtained results.

6 Empirical Illustration

Description of the Sample

Our empirical illustration of the super-efficiency notions draws upon data that are publicly available in the data repository of the Journal of Applied Econometrics5 for the sake of replicability. In particular, we opt for an unbalanced panel of three years of French fruit producers collected by [START_REF] Ivaldi | Comparing Fourier and Translog Specifications of Multiproduct Technology: Evidence from an Incomplete Panel of French Farmers[END_REF] based on annual accounting data collected in a survey.

Two criteria determine the selection of farms: (i) the production of apples is positive, and (ii) the acreage of the orchard is at least five acres. This short panel spans the three successive years from 1984 to 1986. As a description of the technology, one can say that three aggregate inputs deliver two aggregate outputs. The three inputs are: (i) capital (including land), (ii) labor, and (iii) materials.

The two outputs are (i) the production of apples, and (ii) an aggregate of alternative products.

Descriptive statistics for the 405 observations in total and details on the definitions of all variables are available in Appendix 2 in [START_REF] Ivaldi | Comparing Fourier and Translog Specifications of Multiproduct Technology: Evidence from an Incomplete Panel of French Farmers[END_REF]. A striking feature of this sample is the large heterogeneity in terms of size among the different inputs as well as the outputs.

Due to the short length of the panel (just three years), we believe that the use of an intertemporal frontier approach ignoring technical change is justified. Table B.1 presents basic descriptive statistics for the inputs, the outputs, and the input prices: see the Appendix B. One observes basically a lot of heterogeneity and a rather wide range for all inputs and outputs. The range for some of the input prices is smaller.

Empirical Results

Table 1 reports the descriptive statistics of cost efficiency (CE(x p , y p , w | T Λ,Γ ), cost super-efficiency (CE SE (x p , y p , w | T Λ,Γ p ), input-oriented technical efficiency measure (DF i (x p , y p | T Λ,Γ )) and inputoriented technical super-efficiency measure (DF SE i (x p , y p | T Λ,Γ p )) for DMUs using convex and nonconvex technologies under VRS and CRS, respectively. We report the average, the standard deviation, and the minima and maxima (depending on the context) for all these measures. This explains the four horizontal parts of this p ) = 1), respectively. Since the inefficient observations are in common between standard efficiency and super-efficiency concepts, we focus on the distribution of the super-efficiency results that are unity or higher. Therefore, the number of efficient units (#Efficient units), the number of infeasible units (#Infeasible units), as well as the number of observations that we include in the corresponding descriptive statistics computations (#Considered units) is shown in the three last lines of each of the four horizontal part of Table 1. Obviously, for the super-efficiency results the sum of #Infeasible units and #Considered units equals the #Efficient units for the standard efficiency results. Note also that the notion of #Efficient units is not applicable for the super-efficiency results.

Analyzing the results in Table 1, one can draw the following conclusions. First, as expected based on Proposition 5.1(i), the average of DF i (x p , y p | T Λ,V RS ) is larger than the average of CE(x p , y p , w | T Λ,V RS ) under both the convex and nonconvex cases and under both VRS and CRS. Second, Proposition 5.1(iii) cannot be verified at the level of our descriptive statistics: while

we indeed have CE SE (x p , y p , w | T Λ,V RS p ) ≤ DF SE i (x p , y p | T Λ,V RS p
) under both convex and nonconvex cases, under the CRS case this relation does not hold under convexity and nonconvexity. This is due to the fact that the number of observations that we include in the corresponding descriptive statistics is different (see last line with #Considered units). Obviously, the relation in Proposition 5.1(iii) does hold at the level of the common individual observations.6 Third, the minimum of CE SE (x p , y p , w | T Λ,Γ p ) and DF SE i (x p , y p | T Λ,Γ p ) under both the convex and nonconvex cases and under both VRS and CRS is higher than unity. Therefore, both CE SE (x p , y p , w | T Λ,Γ p ) and DF SE i (x p , y p | T Λ,Γ p ) separately yield a unique rank for all feasible observations. Fourth, the obtained VRS results show that 22 units are located on the technically efficient frontier under convexity, among which 7 units are cost efficient, while 185 units are located on the technically efficient frontier under nonconvexity, among which 62 units are cost efficient. Fifth, the results reveal that in the CRS case, 9 units are located on the technically efficient frontier under convexity, whereby 2 units are also cost efficient, while 52 units are located on the technically efficient frontier under nonconvexity, whereby 8 units are cost efficient. Both these conclusions underscore Propositions 5.1(ii) and 5.2: the technical and cost efficiency scores are higher in the nonconvex case.

Fifth, the percentage differences between average nonconvex and convex costs relative to nonconvex costs can be computed by taking the ratio of standard cost efficiency ratios (see first column).7

Under VRS and CRS this amounts to cost difference of 31.00% and 30.95%, respectively.8 This is in the mid range of the cost difference amounts reported in [START_REF] Kerstens | Solution Methods for Nonconvex Free Disposal Hull Models: A Review and Some Critical Comments[END_REF].

We now turn to some basic empirical results of the lower and upper bounds on cost efficiency (i.e., CE(x p , y p , w | T Λ,Γ ) and CE(x p , y p | T Λ,Γ )) and cost super-efficiency (i.e., CE SE (x p , y p ) and CE SE (x p , y p )). Due to space limitations, these empirical results are presented in Appendix D.

We apply the decomposition of cost efficiency (11) and cost super-efficiency (24) to this same empirical example. The convex and nonconvex results for each of these two decompositions are exhibited in The following pertinent conclusions emerge. First, the average of CE(x p , y p , w | T N C,CRS ) and

CE SE (x p , y p , w | T C,CRS p
) are smaller than the average of their decomposition components under both convex and nonconvex technologies. Second, Proposition 5.4 cannot be verified at the level of our descriptive statistics: while the allocative efficiency measure is greater than or equal to the allocative super-efficiency measure under both convex and nonconvex cases, it is not the case that the scale efficiency measure is greater than or equal to the scale super-efficiency measure under both convex and nonconvex cases. Again, this is because the number of observations included in T C,Γ p ) have the highest rank correlation among other efficiency notions. Actually, we compare 20 and 9 units under VRS and CRS, respectively such that only 5 and 2 of these units are cost efficient (see #Considered reported in Table 1). Hence, we obtain high rank correlations only for these observations. Similar results are obtained for the rank correlations between CE(x p , y p , w | T C,Γ ) and CE SE (x p , y p , w | T C,Γ p ) for the nonconvex results (above the diagonal) in both VRS and CRS. Second, comparing convex and nonconvex results on the diagonal under the VRS case, the rank correlations among technical standard and super-efficiency notions is lower than the rank correlations among cost-based standard and super-efficiency notions. Furthermore, the rank correlations among super-efficiency notions are higher than the rank correlations among standard efficiency notions. Third, comparing convex and nonconvex results on the diagonal under the CRS case, ). Fourth, the rank correlations between technical super-efficiency and cost super-efficiency are not significant under the convex case (under the diagonal) for both VRS and CRS. By contrast, the rank correlations between these notions under the nonconvex case (above the diagonal) under both VRS and CRS is significant at the 0.01 and 0.05 levels, respectively.11

We can end with the following overall conclusions. First, the new notion of cost super-efficiency is clearly distinct in terms of ranking from the existing technical super-efficiency concept. This is especially true in the convex case, while in the nonconvex setting the correlations are at least somewhat similar under VRS and CRS in terms of statistical significance. Second, the technical super-efficiency concept is more clearly distinct in terms of ranking than the new cost-based superefficiency concept when comparing convex and nonconvex cases. Furthermore, the distinction in terms of ranking for these super-efficiency concepts between convex and nonconvex cases is largest in the VRS case and smallest in the CRS case.

Conclusions

While the technical super-efficiency concept has been around for about two and a half decades, the cost super-efficiency concept seems entirely new. Apart from the -to our knowledge-two articles treating technical super-efficiency under nonconvexity, we believe we are the first to offer a truly comparative perspective on technical and cost super-efficiency conditioned on this important convexity axiom. In particular, this contribution explores technical and cost super-efficiency from a methodological and empirical perspective conditioning on traditional convex and less common nonconvex technologies and on CRS versus VRS.

After defining all super-efficiency concepts and numerically illustrating some basic issues, we derive a series of theoretical results regarding traditional efficiency concepts as well as super-efficiency concepts. We equally develop some results regarding traditional efficiency and super-efficiency concepts conditional on the axiom of convexity. Finally, we chart the potential infeasibilities governing While the consideration of cost super-efficiency does not change anything to the infeasibility problem for the corresponding technical super-efficiency problem, it is clear that nonconvexity aggravates the infeasibility problem for both the cost super-efficiency and the technical super-efficiency problems. Thus, instead of resolving the infeasibility problem that haunts the super-efficiency literature since its inception, we have deepened this problem. Perhaps, sometimes problems need to be aggravated before these can be successfully resolved.

Among the avenues for future methodological research one can list the following topics. One extension for future research is to analyse the super-efficiency notion in the context of either the revenue function or the profit function. Another avenue is to search for solutions for the infeasibility problem that can be equally applied to the efficiency measures as well as the cost function. Finally, it may be worthwhile to apply these new models in the context of different technical and economic efficiency measures: examples include the directional distance function [START_REF] Chambers | Profit, Directional Distance Functions, and Nerlovian Efficiency[END_REF]), but also the weighted additive model [START_REF] Cooper | Decomposing Profit Inefficiency in DEA Through the Weighted Additive Model[END_REF]) or the slacks-based measure [START_REF] Aparicio | Measuring and Decomposing Profit Inefficiency Through the Slacks-Based Measure[END_REF]).

  first solve model min w∈W {wx k | wx p = 1} for each k ∈ χ(y p ), k = p and then we select the minimum of their optimal values as the lower bound of the cost super-efficiency of DMU p . If we have χ(y p ) = {p}, then there is no lower bound for the cost super-efficiency of DMU p . 4 Numerical Illustration Now we clarify the above Definitions 3.1 and 3.2 by means of Figure 1. The isoquant indicating the combinations of two inputs x 1 and x 2 yielding a given output level y is represented by the polyline ABC and its horizontal and vertical extensions at A and C respectively. This isoquant is the boundary of the input set L(y). We first focus on observation B to illustrate technical and cost efficiency as well as technical and cost super-efficiency measures. The input prices of observation B are such that the cost-minimizing input vector is located at point B on the boundary of the input set L(y): this cost function is plotted as a dotted line. The corresponding technical and cost efficiency measures for point B are DF i (x B , y B | T C,V RS ) = OB OB = 1 and CE(x B , y B , w | T C,V RS ) = OB OB = 1, respectively. Therefore, since DF i (x B , y B | T C,V RS ) = CE(x B , y B , w | T C,V RS ) = 1 this observation B is both technically and cost efficient.

Figure 1 :

 1 Figure 1: Isoquant with combinations of two inputs yielding a given output level y

  cost super-efficiency in Proposition 5.3(i), consider the isoquant of a CRS technology in Figure 2. This isoquant of technology T C,CRS is determined by 6 observations (A, B, D, E, F, G, H) denoted by square dots in the space of two inputs for a given single output. It is represented by the polyline ABDE and its vertical and horizontal extensions at A and E, respectively.

Figure 2 :

 2 Figure 2: Isoquant with combinations of two inputs and single output under convexity and CRS assumptions

Figure 3 illustrates

 3 Figure 3 illustrates Proposition 5.3(ii), namely how infeasibility for super-efficiency under VRS may occur for the case of a single input and a single output under both convex and nonconvex cases. The six observations (A, B, D, E, F, G) determining technologies T C,V RS and T N C,V RS are denoted by square dots. The boundary of technology T C,V RS is represented by the polyline ABD and its vertical and horizontal extensions at A and D, respectively. The boundary of technology T N C,V RS is represented by the polyline AHBID and its vertical and horizontal extensions at A and D, respectively.

Figure 3 :

 3 Figure 3: Convex and nonconvex technologies with single input and single output under VRS assumption

  Hence, observation D is technically and cost efficient under both convexity and nonconvexity. Removing observation D alters technology in the convex case to the new technology T C,V RS D represented by the polyline ABE and its vertical and horizontal extensions at A and E, respectively. Similarly, removal of observation D changes technology in the nonconvex case to the new technology T N C,V RS D represented by the polyline AHBJE and its vertical and horizontal extensions at A and E, respectively. Clearly, there are no observations in technologies T C,V RS D and T N C,V RS D with at least current output level of observation D. Therefore, both technical and cost super-efficiency measures under VRS and under convexity or nonconvexity are infeasible.

Figure 4 illustrates

 4 Figure4illustrates Proposition 5.3(iii), namely if an observation results in an infeasibility under convexity, then this same observation also leads to an infeasibility under nonconvexity for the technical super-efficiency measure. Also, it shows that the inverse of Proposition 5.3(iii) is not satisfied: i.e., if an observation results in an infeasibility under nonconvexity, then this same observation may lead to a feasibility under convexity for the technical super-efficiency measure.

Figure 4 :

 4 Figure 4: Convex and nonconvex technologies with single input and two outputs under VRS assumption

  the rank correlations are remarkably high overall among efficiency notions, and these are highest for CE(x p , y p , w | T Λ,CRS ) and CE SE (x p , y p , w | T Λ,CRS p ) compared to DF i (x p , y p | T Λ,CRS ) and DF SE i (x p , y p | T Λ,CRS p

  these super-efficiency concepts. Most of our theoretical results are new to the frontier literature and even broaden our knowledge about the basic properties of the cost function within the context of incentive-based regulation theory employing frontier methodologies. An empirical section based on secondary data serves to illustrate some of these theoretical results. One main lesson is that the cost super-efficiency notion yields different rankings from the technical super-efficiency concept. Another lesson is that both cost and technical super-efficiency notions rank differently under convex and nonconvex technologies. The latter lesson underscores the differences between convex and nonconvex costs already listed in[START_REF] Kerstens | Solution Methods for Nonconvex Free Disposal Hull Models: A Review and Some Critical Comments[END_REF] and the substantial effect of convexity on technical efficiency and cost function based capacity concepts documented in[START_REF] Kerstens | Convex and Nonconvex Input-Oriented Technical and Economic Capacity Measures: An Empirical Comparison[END_REF].

  ). Consequently, observation (x p , y p ) is technically inefficient by the conventional model (2) if and only if the super-efficiency estimate DF SE i (x p , y p | T Λ,Γ p ) < 1 also indicates inefficiency. This observation is technically efficient by the conventional model (2) if and only if DF SE i (x p , y p | T Λ,Γ p ) ≥ 1: it indicates efficiency when DF SE i (x p , y p | T Λ,Γ p ) = 1 and it indicates super-efficiency when DF SE i (x p , y p

Table 1 :

 1 the first two parts report VRS results, while the last two parts list CRS results, and each of these two major parts reports results under convexity and nonconvexity, respectively.Note that in the VRS case for 2 and 8 observations under convexity and nonconvexity respectively, the corresponding modelsDF SE i (x p , y p | T Λ,V RS p ) and CE SE (x p , y p , w | T Λ,V RS p) are infeasible. This illustrates that the inverses of Proposition 5.3(iii) and (iv) are not satisfied. As a result, the prevalence of infeasibility is higher in the nonconvex case. Therefore, we do not include these infeasible observations in the corresponding descriptive statistics computations under VRS. By contrast, no observations are infeasible in the CRS case: this is a clear illustration of Proposition 5.3(i).

Table 1 :

 1 Descriptive Statistics for Technical and Cost Efficiency and Super-Efficiency under VRS and CRS and in both Convex and Nonconvex Cases VRS, Convex CE(x p , y p, w | T C,V RS ) CE SE (x p , y p , w | T C,V RS Nonconvex CE(x p , y p , w | T N C,V RS ) CE SE (x p , y p , w | T N C,V RS p ) DF i (x p , y p | T N C,V RS ) DF SE i (x p , y p | T N C,V RS Nonconvex CE(x p , y p , w | T N C,CRS ) CE SE (x p , y p , w | T N C,CRS p ) DF i (x p , y p | T N C,CRS ) DF SE i (x p , y p | T N C,CRSFurthermore, to compute the descriptive statistics for the super-efficiency results (i.e., CE SE (x p , y p , w | T Λ,Γ p ) and DF SE i (x p , y p | T Λ,Γ p )), we only consider the observations that are cost efficient (i.e., CE(x p , y p , w | T Λ,Γ p ) = 1) and input-oriented technically efficient (i.e., DF i (x p , y p | T Λ,Γ

	p	)	DF i (x p , y p | T C,V RS )	DF SE i (x p , y p | T C,V RS p	)

Table 2 .

 2 This explains the four horizontal parts of this table. As mentioned in subsection 3.3, if observation (x p , y p ) is inefficient under VRS (i.e., DF i (x p , y p | T Λ,V RS ) < 1), then its decomposition of overall or cost efficiency (11) and its decomposition of overall or cost super-efficiency (24) yield the same results. Hence, to report the descriptive statistics for the components of the decomposition of cost super-efficiency (24), we only consider the observations that are input-oriented technically efficient. The number of observations that we include in the corresponding descriptive statistics computations (#Considered units) is shown in the last line of every part of Table2. Note that this #Considered units for the decomposition of cost super-efficiency corresponds to the #Considered units for the VRS technical super-efficiency results in Table1.

Table 2 :

 2 Descriptive Statistics for Overall Efficiency and Overall Super-Efficiency Decomposition

				Decomposition of cost efficiency
	Convex CE(x Average	0.261			0.604	0.612	0.722
	Stand. Dev.		0.161			0.193	0.259	0.177
	Min		0.036			0.187	0.073	0.242
	Max		1.000			1.000	1.000	1.000
	#Considered units		405			405	405	405
	Nonconvex CE(x Average	0.378			0.847	0.664	0.659
	Stand. Dev.		0.219			0.190	0.238	0.180
	Min		0.039			0.359	0.079	0.253
	Max		1.000			1.000	1.000	1.000
	#Considered units		405			405	405	405
			Decomposition of cost super-efficiency
	Convex	CE SE (x p , y p , w | T C,CRS p	)	DF SE i (x p	)
	Average		0.441			1.251	0.644	0.558
	Stand. Dev.		0.314			0.169	0.289	0.228
	Min		0.052			1.033	0.066	0.247
	Max		1.303			1.661	0.999	0.892
	#Considered units		20			20	20	20
	Nonconvex	CE SE (x p , y p , w | T N C,CRS p	) DF SE i (x p , y p | T N C,V RS	p	)
	Average		0.526			1.456	0.614	0.610
	Stand. Dev.		0.253			0.488	0.165	0.188
	Min		0.081			1.002	0.111	0.186
	Max		1.724			3.513	0.982	0.978
	#Considered units		177			177	177	177

p , y p , w | T C,CRS ) DF i (x p , y p | T C,V RS ) SCE(x p , y p | T C,. ) AE(x p , y p , w | T C,CRS ) p , y p , w | T N C,CRS ) DF i (x p , y p | T N C,V RS ) SCE(x p , y p | T N C,. ) AE(x p , y p , w | T N C,CRS ) p , y p | T C,V RS p ) SCE SE (x p , y p | T C,. p ) AE SE (x p , y p , w | T C,CRS p ) SCE SE (x p , y p | T N C,. p ) AE SE (x p , y p , w | T N C,CRS

Note that the convex VRS technology does not satisfy inaction.

The infeasibility of the super-efficiency models and of the Luenberger productivity indicator is structurally similar: both consist in one (or more) observation(s) being situated outside a technology without there being a guarantee that the distance from the outside point to this technology can be achieved. One conjecture is that a solution to the infeasibility of both models could be structurally similar.

Web site: http://qed.econ.queensu.ca/jae/

These results at the level of the individual observations are available upon request.

In particular, taking a ratio of cost efficiency ratios based on cost frontier estimates under convexity and nonconvexity nets out the observed cost and reveals the difference in cost estimates under convexity and nonconvexity.

8 Note that 31.00% = (1-0.434/0.629) and 30.95% = (1-0.261/0.378). Note also that under CRS the cost efficiency ratios under convexity and nonconvexity of 0.261 and 0.378 have earlier been reported in the last column of Table5in[START_REF] Kerstens | Convex and Nonconvex Input-Oriented Technical and Economic Capacity Measures: An Empirical Comparison[END_REF].

These results at the level of the individual observations are available upon request.

Note that dependency is intrinsic to nonparametric frontier estimators: i.e., technical and cost efficiency and superefficiency levels depend on sample size, among others. Matlab code developed by P.J. Kerstens based on Li, Maasoumi, and Racine (2009) is found at: https://github.com/kepiej/DEAUtils.

Following the suggestion of a referee, we also report the Kruskal-Wallis rank test to determine differences in rank among groups of efficiency components. For reasons of space, these statistics are reported in Appendix C.

the corresponding descriptive statistics is different (see last line with #Considered units). But, of course this relation in Proposition 5.4 holds true at the level of the common individual observations. 9

Third, the results show that the average of allocative super-efficiency is smaller than the average of scale super-efficiency, under both convex and nonconvex cases. Fourth, the average of scale efficiency under nonconvexity is higher than under convexity, while the average of scale super-efficiency under convexity is higher than under nonconvexity. Fifth, the average of allocative efficiency under convexity is higher than under nonconvexity while the average on allocative super-efficiency under convexity is smaller than under nonconvexity.

To formally assess the above reported differences in distributions, we make use of a nonparametric test proposed initially by [START_REF] Li | Nonparametric Testing of Closeness between Two Unknown Distribution Functions[END_REF] and further refined by [START_REF] Fan | On Goodness-of-fit Tests for Weakly Dependent Processes Using Kernel Method[END_REF] and others. The most recent development is probably by [START_REF] Li | A Nonparametric Test for Equality of Distributions with Mixed Categorical and Continuous Data[END_REF]. This nonparametric test focuses on the differences between entire distributions instead of looking at, for instance, first moments (e.g., the Wilcoxon signed-ranks test). It tests for the eventual statistical significance of differences between two kernel-based estimates of density functions f and g of a random variable x. The null hypothesis states the equality of both density functions almost everywhere: H 0 : f (x) = g(x) for all x. By contrast, the alternative hypothesis negates this equality of both density functions:

for some x. This test is valid for both dependent and independent variables. 10 Table 3 reports the Li-test results among all technical and cost efficiency as well as technical and cost super-efficiency notions under VRS and CRS, respectively. Both VRS and CRS parts of this Table are structured as follows. First, components on the diagonal (in bold) depict the Li-test statistic between the convex and nonconvex cases. Second, the components under the diagonal show the Li-test statistic between convex efficiency measures, and the components above the diagonal show the Li-test statistic between nonconvex efficiency notions.

Table 3: Li-Test Among All Technical and Cost Efficiency as well as Technical and Cost Super-Efficiency Notions under VRS and CRS CRS parts, the components on the diagonal show the rank correlations between convex and nonconvex cases. The components under the diagonal show the rank correlations between convex efficiency notions, and the components above the diagonal show the rank correlations between nonconvex efficiency notions. Note that the numbers in the parentheses indicate the number of observations in the computations of the Spearman rank correlation coefficients between the corresponding components.

To obtain the rank correlations between convex and nonconvex cases on the diagonals, we consider all feasible observations. To be precise, by solving the super-efficiency model ( 19) under VRS, we obtain 403 feasible observations under convexity, but only 397 of these are feasible under nonconvexity. Hence, we consider only these 397 units in common to obtain the rank correlations between convex and nonconvex cases. Note that in the CRS case all 405 units are feasible and are included in the computations.

To obtain the rank correlations between convex (under the diagonal) and nonconvex (above the diagonal) efficiency notions under both VRS and CRS respectively, we only consider the inputoriented technically efficient observations. Note that the numbers 20 and 177 for the VRS case and the numbers 9 and 52 for the CRS case coincide with the #Considered reported in Table 1. Furthermore, since for all input-oriented technically efficient observations we obviously have DF i (x p , y p | T Λ,Γ ) = 1, we cannot compute Spearman rank correlations between DF i (x p , y p | T Λ,Γ ) and the other components under both VRS and CRS.