Skip to main content
SpringerLink
Log in

Piecewise Frontier Analysis

  • Chapter
  • First Online:
Productivity and Efficiency Analysis

  • 2072 Accesses

Abstract

Estimating/predicting levels of efficiency involves estimating production frontiers. A widely-used estimation approach involves enveloping scatterplots of data points as tightly as possible without violating any assumptions that have been made about production technologies. Some of the most common assumptions lead to estimated frontiers that are comprised of multiple linear segments (or pieces). The associated frontiers are known as piecewise frontiers. This chapter explains how to estimate the unknown parameters of so-called piecewise frontier models (PFMs). It then explains how the estimated parameters can be used to analyse efficiency and productivity change. The focus is on data envelopment analysis (DEA) estimators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Färe et al. (1990) consider the case where a subset of inputs are pre-determined (or ‘fixed’). The fixed-input constraint in their so-called short-run profit maximisation problem plays the same role as, but has a different interpretation to, the environmental variable constraint in problem (6.13).

  2. 2.

    This estimation problem is obtained by replacing q and x in problem (6.15) with \(\rho q_{it}\) and \(\delta x_{it}\), then using the linear homogeneity properties of the aggregator functions to simplify. For example, the constraint \(X(\delta x_{it})=1\) implies that \(\delta = 1/X(x_{it})\), which in turn implies that \(\delta x_{it}=x_{it}/X(x_{it})\).

  3. 3.

    In this context, ‘consistent’ means that the sampling distributions of the estimators collapse to the true levels of (in)efficiency as the numbers of firms used to estimate production frontiers become infinitely large.

  4. 4.

    This formula is based on the result that if \(u_{it}\) is an independent exponential random variable with scale parameter \(\sigma _t >0\), then \(P[-\sigma _t \ln (1-\alpha /2)\le u_{it} \le -\sigma _t \ln (\alpha /2)]=1-\alpha \Rightarrow P[ (\alpha /2)^{\sigma _t} \le \exp (-u_{it}) \le (1-\alpha /2)^{\sigma _t}]=1-\alpha \).

  5. 5.

    Consider an F distribution with the numerator degrees of freedom equal to the denominator degrees of freedom. A half-F distribution is such a distribution truncated from below at 1. The critical value that leaves an area of \(\alpha \) in the right-hand tail of a half-F distribution is the value that leaves an area of \(\alpha /2\) in the right-hand tail of the corresponding untruncated F distribution.

  6. 6.

    Primal and dual indices are only proper indices if the parameters of distance, revenue and cost functions do not vary across observations. Except in restrictive special cases (e.g., there is only one input and one output, there is no environmental change, and production frontiers exhibit CRS), PFMs are underpinned by functions with parameters that do vary across observations.

  7. 7.

    Additive TFPI numbers can also be computed using estimated representative normalised shadow prices as weights. The DPIN software of O’Donnell (2010a) computes so-called ‘Färe-Primont’ TFPI numbers in this way. If average estimated normalised shadow prices are proportional to average observed prices, then the associated additive index numbers are equal to Lowe index numbers.

  8. 8.

    If average estimated shadow value shares are equal to average observed value shares, then the associated multiplicative index numbers are equal to geometric Young index numbers.

References

  • Afsharian M, Ahn H (2015) The overall Malmquist index: a new approach for measuring productivity changes over time. Ann Oper Re 226(1):1–27

    Article  Google Scholar 

  • Afsharian M, Podinovski V (2018) A linear programming approach to efficiency evaluation in nonconvex metatechnologies. Eur J Oper Res 268(1):268–280

    Article  Google Scholar 

  • Allen R, Athanassopoulos A, Dyson R, Thanassoulis E (1997) Weight restrictions and value judgements in data envelopment analysis: evolution, development and future directions. Ann Oper Res 73:13–34

    Article  Google Scholar 

  • Anik AR, Rahman S, Sarker J (2017) Agricultural productivity growth and the role of capital in South Asia (1980–2013). Sustainability 9(3):470

    Article  Google Scholar 

  • Baležentis T (2015) The sources of the total factor productivity growth in Lithuanian family farms: A Färe-Primont index approach. Prague Econ Papers 24(2):225–241

    Article  Google Scholar 

  • Banker R (1993) Maximum likelihood, consistency and data envelopment analysis: a statistical foundation. Manag Sci 39(10):1265–1273

    Article  Google Scholar 

  • Banker R (1996) Hypothesis tests using data envelopment analysis. J Prod Anal 7(2–3):139–159

    Article  Google Scholar 

  • Banker R, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW, Schinnar AP (1981) A bi-extremal principle for frontier estimation and efficiency evaluations. Manag Sci 27(12):1370–1382

    Article  Google Scholar 

  • Baráth L, Fertö I (2017) Productivity and convergence in European agriculture. J Agri Econ 68(1):228–248

    Article  Google Scholar 

  • Battese G, Rao D, O’Donnell C (2004) A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. J Prod Anal 21(1):91–103

    Article  Google Scholar 

  • Beltran-Esteve M, Gomez-Limon J, Picazo-Tadeo A, Reig-Martinez E (2014) A metafrontier directional distance function approach to assessing eco-efficiency. J Prod Anal 41:69–83

    Article  Google Scholar 

  • Bogetoft P, Otto L (2015) Benchmarking: benchmark and frontier analysis using DEA and SFA. R package version 2015-7-8

    Google Scholar 

  • Boles J (1966) Efficiency squared-efficient computation of efficiency indexes. In: Thirty Ninth annual meeting of the Western farm economics association, pp 137–142

    Google Scholar 

  • Brännlund R, Färe R, Grosskopf S (1995) Environmental regulation and profitability: an application to Swedish pulp and paper mills. Environ Resour Econ 6:23–36

    Article  Google Scholar 

  • Brännlund R, Chung Y, Färe R, Grosskopf S (1998) Emissions trading and profitability: the Swedish pulp and paper industry. Environ Resour Econ 12:345–356

    Article  Google Scholar 

  • Bressler R (1966) The measurement of productive efficiency. In: Thirty Ninth annual meeting of the Western farm economics association, pp 129–136

    Google Scholar 

  • Briec W, Kerstens K, Van de Woestyne I (2016) Congestion in production correspondences. J Econ 119(1):65–90

    Google Scholar 

  • Briec W, Kerstens K, Van de Woestyne I (2018) Hypercongestion in production correspondences: an empirical exploration. Appl Econ 50(27):2938–2956

    Article  Google Scholar 

  • Burley H (1980) Productive efficiency in U.S. manufacturing: a linear programming approach. Rev Econ Stat 62(4):619–622

    Article  Google Scholar 

  • Carrington R, Coelli T, Rao D (2005) The performance of Australian universities: conceptual issues and preliminary results. Econ Papers 24(2):145–163

    Article  Google Scholar 

  • Carrington R, O’Donnell C, Rao D (2016) Australian university productivity growth and public funding revisited. Stud High Educ 1–22: https://doi.org/10.1080/03075,079.2016.1259,306

  • Casu B, Ferrari A, Zhao T (2013) Regulatory reform and productivity change in Indian banking. Rev Econ Stat 95(3):1066–1077

    Article  Google Scholar 

  • Charnes A, Cooper W (1962) Programming with linear fractional functionals. Naval Res Logistics Q 9(3–4):181–186

    Article  Google Scholar 

  • Charnes A, Cooper W, Rhodes E (1978) Measuring the efficiency of decision-making unit. Eur J Oper Res 2(6):429–444 (see also “Corrections” op.cit. 3(4):339)

    Article  Google Scholar 

  • Charnes A, Cooper W, Rhodes E (1981) Evaluating program and managerial efficiency: an application of data envelopment analysis to Program follow through. Manag Sci 27(6):668–697

    Article  Google Scholar 

  • Chattopadhyay S, Heffley D (1994) Are for-profit nursing homes more efficient? Data envelopment analysis with a case-mix constraint. East Econ J 20(2):171–186

    Google Scholar 

  • Coelli T, Grifell-Tatje E, Perelman S (2002) Capacity utilisation and profitability: a decomposition of short-run profit efficiency. Int J Prod Econ 79(3):261–278

    Article  Google Scholar 

  • Coelli T, Rao D, O’Donnell C, Battese G (2005) An introduction to efficiency and productivity analysis, 2nd edn. Springer, New York

    Google Scholar 

  • Cooper W, Seiford L, Zhu J (2000) A unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA. Socio-Econ Plann Sci 34(1):1–25

    Article  Google Scholar 

  • Cooper W, Seiford L, Zhu J (2001) Slacks and congestion: response to a comment by R. Fare and S. Grosskopf. Socio-Econ Plann Sci 35(3):205–215

    Google Scholar 

  • Curi C, Guarda P, Lozano-Vivas A, Zelenyuk V (2013) Is foreign-bank efficiency in financial centers driven by home or host country characteristics? J Prod Anal 40:367–385

    Article  Google Scholar 

  • Demchuk P, Zelenyuk V (2009) Testing differences in efficiency of regions within a country: the case of Ukraine. J Prod Anal 32(2):81–102

    Article  Google Scholar 

  • Deprins D, Simar L, Tulkens H (1984) Measuring labor efficiency in post offices. In: Marchand M, Pestieau P, Tulkens H (eds) The performance of public enterprises: concepts and measurements. North Holland Publishing Company, Amsterdam

    Google Scholar 

  • Fan Y, Li Q, Weersink A (1996) Semiparametric estimation of stochastic production frontier models. J Bus Econ Stat 14(4):460–468

    Google Scholar 

  • Färe R, Grosskopf S (1998) Congestion: a note. Socio-Econ Plann Sci 32(1):21–23

    Article  Google Scholar 

  • Färe R, Grosskopf S (2000) Slacks and congestion: a comment. Socio-Econ Plann Sci 34(1):27–33

    Article  Google Scholar 

  • Färe R, Grosskopf S (2001) When can slacks be used to identify congestion? An answer to W. W. Cooper, L. Seiford and J. Zhu. Socio-Econ Plann Sci 35(3):217–221

    Google Scholar 

  • Färe R, Grosskopf S, Logan J (1983) The relative efficiency of Illinois electric utilities. Resour Energy 5(4):349–367

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lovell C (1985) The measurement of efficiency of production. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Färe R, Grosskopf S, Kokkelenberg E (1989) Measuring plant capacity, utilization and technical change: a non-parametric approach. Int Econ Rev 30(3):655–666

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lee H (1990) A nonparametric approach to expenditure constrained profit maximization. Am J Agri Econ 72(3):574–581

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lovell C (1994) Production frontiers. Cambridge University Press, Cambridge

    Google Scholar 

  • Färe R, Grosskopf S, Norris M, Zhang Z (1994) Productivity growth, technical progress, and efficiency change in industrialized countries. Am Econ Rev 84(1):66–83

    Google Scholar 

  • Färe R, Grosskopf S, Lee WF (2001) Productivity and technical change: the case of Taiwan. Appl Econ 33(15):1911–1925

    Article  Google Scholar 

  • Färe R, Grosskopf S, Hernandez-Sancho F (2004a) Environmental performance: an index number approach. Resour Energy Econ 26(4):343–352

    Article  Google Scholar 

  • Färe R, Grosskopf S, Weber W (2004b) The effect of risk-based capital requirements on profit efficiency in banking. Appl Econ 36(15):1–13

    Article  Google Scholar 

  • Farrell M (1957) The measurement of productive efficiency. J R Stat Soc Ser A (General) 120(3):253–290

    Article  Google Scholar 

  • Farrell M, Fieldhouse M (1962) Estimating efficient production functions under increasing returns to scale. J R Stat Soc Ser A (General) 125(2):252–267

    Article  Google Scholar 

  • Forsund F, Sarafoglou N (2002) The origins of data envelopment analysis. J Prod Anal 17(1):23–40

    Article  Google Scholar 

  • Gijbels I, Mammen E, Park B, Simar L (1999) On estimation of monotone and concave frontier functions. J Am Stat Assoc 94(445):220–228

    Article  Google Scholar 

  • Grosskopf S (1996) Statistical inference and nonparametric efficiency: a selective survey. J Prod Anal 7:161–176

    Article  Google Scholar 

  • Hoff A (2007) Second stage DEA: comparison of approaches for modelling the DEA score. Eur J Oper Res 181(1):425–435

    Article  Google Scholar 

  • Huang C, Fu TT (1999) An average derivative estimation of stochastic frontier. J Prod Anal 12:45–53

    Article  Google Scholar 

  • Huang MY, Juo JC, Fu TT (2015) Metafrontier cost Malmquist productivity index: an application to Taiwanese and Chinese commercial banks. J Prod Anal 44:321–335

    Article  Google Scholar 

  • Islam N, Xayavong V, Kingwell R (2014) Broadacre farm productivity and profitability in south-western Australia. Aust J Agri Resour Econ 58(2):147–170

    Article  Google Scholar 

  • Johnes J, Izzeldin M, Pappas V (2014) A comparison of performance of Islamic and conventional banks 2004–2009. J Econ Behav Organ 103(Supplement):S93–S107

    Article  Google Scholar 

  • Johnson A, Kuosmanen T (2011) One-stage estimation of the effects of operational conditions and practices on productive performance: asymptotically normal and efficient, root-n consistent StoNEZD method. J Prod Anal 36(2):219–230

    Article  Google Scholar 

  • Johnson A, Kuosmanen T (2012) One-stage and two-stage DEA estimation of the effects of contextual variables. Eur J Oper Res 220(2):559–570

    Article  Google Scholar 

  • Kerstens K, Vanden Eeckaut P (1999) Estimating returns to scale using non-parametric deterministic technologies: a new method based on goodness-of-fit. Eur J Oper Res 113(1):206–214

    Article  Google Scholar 

  • Kerstens K, O’Donnell C, Van de Woestyne I (2015) Frontier metatechnologies and convexity: a restatement. Document de travail du LEM 2015-08, Universit’e de Lille

    Google Scholar 

  • Khan F, Salim R, Bloch H (2015) Nonparametric estimates of productivity and efficiency change in Australian broadacre agriculture. Aust J Agri Resour Econ 59(3):393–411

    Article  Google Scholar 

  • Kneip A, Park B, Simar L (1998) A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory 14(6):783–793

    Article  Google Scholar 

  • Kneip A, Simar L, Wilson P (2008) Asymptotics and consistent bootstraps for DEA estimators in nonparametric frontier models. Econometric Theory 24:1663–1697

    Article  Google Scholar 

  • Kneip A, Simar L, Wilson P (2011) A computationally efficient, consistent bootstrap for inference with non-parametric DEA estimators. Comput Econ 38(4):483–515

    Article  Google Scholar 

  • Kneip A, Simar L, Wilson P (2015) When bias kills the variance: central limit theorems for DEA and FDH efficiency scores. Econometric Theory 31(2):394–422

    Article  Google Scholar 

  • Korostelev A, Simar L, Tsybakov A (1995) Efficient estimation of monotone boundaries. Ann Stat 23(2):476–489

    Article  Google Scholar 

  • Koutsomanoli-Filippaki A, Margaritis D, Staikouras C (2012) Profit efficiency in the European Union banking industry: a directional technology distance function approach. J Prod Anal 37:277–293

    Article  Google Scholar 

  • Kumar S, Russell R (2002) Technological change, technological catch-up, and capital deepening: relative contributions to growth and convergence. Am Econ Rev 92(3):527–548

    Article  Google Scholar 

  • Kumbhakar S, Park B, Simar L, Tsionas E (2007) Nonparametric stochastic frontiers: a local maximum likelihood approach. J Econometrics 137(1):1–27

    Article  Google Scholar 

  • Kumtong R, Saosaovaphak A, Chaiboonsri C (2017) Measuring the total factor productivity for international seaport in South East Asia (ASEAN). J Adv Res Soc Sci Humanit 2(3):154–167

    Google Scholar 

  • Kuosmanen T, Kortelainen M (2012) Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints. J Prod Anal 38(1):11–28

    Article  Google Scholar 

  • Kuosmanen T, Sipiläinen T (2009) Exact decomposition of the Fisher ideal total factor productivity index. J Prod Anal 31(3):137–150

    Article  Google Scholar 

  • Laurenceson J, O’Donnell CJ (2014) New estimates and a decomposition of provincial productivity change in China. China Econ Rev 30:86–97

    Article  Google Scholar 

  • Li Q (1996) Nonparametric testing of closeness between two unknown distribution functions. Econometric Rev 15(3):261–274

    Article  Google Scholar 

  • Li Q (1999) Nonparametric testing the similarity of two unknown density functions: local power and bootstrap analysis. J Nonparametric Stat 11:189–213

    Article  Google Scholar 

  • McDonald J (2009) Using least squares and tobit in second stage DEA efficiency analyses. Eur J Oper Res 197(2):792–798

    Article  Google Scholar 

  • Milner C, Weyman-Jones T (2003) Relative national efficiency and country size: evidence for developing countries. Rev Dev Econ 7(1):1–14

    Article  Google Scholar 

  • Mohammad A (2015) Analysis of total factor productivity change among Indian commercial banks using Fare-Primont index. Asian J Res Banking Finance 5(2):183–193

    Article  Google Scholar 

  • Mugera A, Langemeier M, Ojede A (2016) Contributions of productivity and relative price change to farm-level profitability change. Am J Agri Econ 98(4):1210–1229

    Article  Google Scholar 

  • Nguyen PA, Simioni M (2015) Productivity and efficiency of Vietnamese banking system: new evidence using Färe-Primont index analysis. Appl Econ 47(41):4395–4407

    Article  Google Scholar 

  • O’Donnell C (2010a) DPIN version 1.0: A program for decomposing productivity index numbers. Centre for efficiency and productivity analysis working papers WP01/2010, University of Queensland

    Google Scholar 

  • O’Donnell C (2010b) Measuring and decomposing agricultural productivity and profitability change. Aust J Agri Resour Econ 54(4):527–560

    Article  Google Scholar 

  • O’Donnell C (2012) Nonparametric estimates of the components of productivity and profitability change in U.S. agriculture. Am J Agri Econ 94(4):873–890

    Article  Google Scholar 

  • O’Donnell C, Nguyen K (2013) An econometric approach to estimating support prices and measures of productivity change in public hospitals. J Prod Anal 40(3):323–335

    Article  Google Scholar 

  • O’Donnell C, Rao D, Battese G (2008) Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Econ 34(2):231–255

    Article  Google Scholar 

  • O’Donnell C, Fallah-Fini S, Triantis K (2017) Measuring and analysing productivity change in a metafrontier framework. J Prod Anal 47(2):117–128

    Article  Google Scholar 

  • Park B, Simar L, Weiner C (2000) The FDH estimator for productivity efficiency scores. Econometric Theory 16(6):855–877

    Article  Google Scholar 

  • Pastor J, Asmild M, Lovell C (2011) The biennial Malmquist productivity change index. Socio-Econ Plann Sci 45(1):10–15

    Article  Google Scholar 

  • Podinovski V (2004a) Production trade-offs and weight restrictions in data envelopment analysis. J Oper Res Soc 55(12):1311–1322

    Article  Google Scholar 

  • Podinovski V (2004b) Suitability and redundancy of non-homogeneous weight restrictions for measuring the relative efficiency in DEA. Eur J Oper Res 154:380–395

    Article  Google Scholar 

  • Podinovski V (2007) Computation of efficient targets in DEA models with production trade-offs and weight restrictions. Eur J Oper Res 181(2):586–591

    Article  Google Scholar 

  • Podinovski V, Bouzdine-Chameeva T (2013) Weight restrictions and free production in data envelopment analysis. Oper Res 61(2):426–437

    Article  Google Scholar 

  • Portela M, Thanassoulis E (2005) Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. Eur J Oper Res 162(3):850–866

    Article  Google Scholar 

  • Rahman S, Salim R (2013) Six decades of total factor productivity change and sources of growth in Bangladesh agriculture (1948–2008). J Agri Econ 64(2):275–294

    Article  Google Scholar 

  • Ray S (1988) Data envelopment analysis: nondiscretionary inputs and efficiency: an alternative interpretation. Socio-Econ Plann Sci 22(4):167–176

    Article  Google Scholar 

  • Seiford L, Thrall R (1990) Recent developments in DEA: the mathematical programming approach to frontier analysis. J Econometrics 46(1–2):7–38

    Article  Google Scholar 

  • Seitz W (1966) Efficiency measures for steam-electric generating plants. In: Thirty Ninth annual meeting of the Western farm economics association, pp 143–151

    Google Scholar 

  • Shiu A, Zelenyuk V (2012) Production efficiency versus ownership. In: van Keilegom I, Wilson P (eds) Exploring research frontiers in contemporary statistics and econometrics. Springer, chap 2, pp 23–44

    Google Scholar 

  • Simar L, Wilson P (1998) Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Manag Sci 44(1):49–61

    Article  Google Scholar 

  • Simar L, Wilson P (2000) Statistical inference in nonparametric frontier models: the state of the art. J Prod Anal 13(1):49–78

    Article  Google Scholar 

  • Simar L, Wilson P (2007) Estimation and inference in two-stage, semi-parametric models of production processes. J Econometrics 136:31–64

    Article  Google Scholar 

  • Simar L, Wilson P (2011a) Inference by the m out of n bootstrap in nonparametric frontier models. J Prod Anal 36(1):33–53

    Article  Google Scholar 

  • Simar L, Wilson P (2011b) Two-stage DEA: Caveat emptor. J Prod Anal 36(2):205–218

    Article  Google Scholar 

  • Simar L, Wilson P (2013) Estimation and inference in nonparametric frontier models: recent developments and perspectives. Found Trends Econometrics 5(3–4):183–337

    Google Scholar 

  • Simar L, Wilson P (2015) Statistical approaches for non-parametric frontier models: a guided tour. Int Stat Rev 83(1):77–110

    Article  Google Scholar 

  • Simar L, Zelenyuk V (2006) On testing equality of distributions of technical efficiency scores. Econometric Rev 25(4):497–522

    Article  Google Scholar 

  • Sitorus B (1966) Productive efficiency and redundant factors of production in traditional agriculture of underdeveloped countries. In: Thirty Ninth annual meeting of the Western farm economics association, pp 153–158

    Google Scholar 

  • Suhariyanto K, Thirtle C (2001) Asian agricultural productivity and convergence. J Agri Econ 52(3):96–110

    Article  Google Scholar 

  • Temoso O, Villano R, Hadley D (2015) Agricultural productivity, efficiency and growth in a semi-arid country: a case study of Botswana. Afr J Agri Resour Econ 10(3):192–206

    Google Scholar 

  • Tozer P, Villano R (2013) Decomposing productivity and efficiency among Western Australian grain producers. J Agri Resour Econ 38(3):312–326

    Google Scholar 

  • Tulkens H, Vanden Eeckaut P (1995) Non-parametric efficiency, progress and regress measures for panel data: methodological aspects. Eur J Oper Res 80:474–499

    Article  Google Scholar 

  • Zelenyuk V (2009) Power of significance test of dummy variables in two-stage efficiency analysis model. Appl Econ Lett 16(15):1493–1495

    Article  Google Scholar 

  • Zhang N, Zhou P, Choi Y (2013) Energy efficiency, CO\(_2\) emission performance and technology gaps in fossil fuel electricity generation in Korea: a meta-frontier non-radial directional distance function analysis. Energy Policy 56(May):653–662

    Article  Google Scholar 

  • Zhu J (2009) Quantitative models for performance evaluation and benchmarking: data envelopment analysis with spreadsheets, 2nd edn. Springer

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Queensland, Brisbane, QLD, Australia

    Christopher J. O’Donnell

Authors
  1. Christopher J. O’Donnell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher J. O’Donnell .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

O’Donnell, C.J. (2018). Piecewise Frontier Analysis. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-2984-5_6

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-2982-1

  • Online ISBN: 978-981-13-2984-5

  • eBook Packages: Economics and FinanceEconomics and Finance (R0)

Publish with us

Policies and ethics

Search

Navigation