IDEAS home Printed from https://ideas.repec.org/p/tse/wpaper/23435.html
   My bibliography  Save this paper

Probabilistic Characterization of Directional Distances and their Robust Versions

Author

Listed:
  • Simar, Léopold
  • Vanhems, Anne

Abstract

In productivity analysis, the performance of production units is measured through the distance of the individual decision making units (DMU) to the technology which is defined as the frontier of the production set. Most of the existing methods, Farrell-Debreu and Shephard radial measures (input or output oriented) and hyperbolic distance functions, rely on multiplicative measures of the distance and so require to deal with strictly positive inputs and outputs. This can be critical when the data contain zero or negative values as in financial data bases for the measure of funds performances. Directional distance function is an alternative that can be viewed as an additive measure of efficiency. We show in this paper that using a probabilistic formulation of the production process, the directional distance can be expressed as simple radial or hyperbolic distance up to a simple transformation of the inputs/outputs space. This allows to propose simple methods of estimation but also to transfer easily most of the known properties of the estimators shared by the radial and hyperbolic distances. In addition, the formulation allows to define robust directional distances in the lines of alpha-quantile or order-m partial frontiers. Finally we can also define conditional directional distance functions, conditional to environmental factors. To illustrate the methodology, we show how it can be implemented using a Mutual Funds database.

Suggested Citation

  • Simar, Léopold & Vanhems, Anne, 2010. "Probabilistic Characterization of Directional Distances and their Robust Versions," TSE Working Papers 10-195, Toulouse School of Economics (TSE).
  • Handle: RePEc:tse:wpaper:23435
    as

    Download full text from publisher

    File URL: http://www.tse-fr.eu/sites/default/files/medias/doc/wp/etrie/10-195.pdf
    File Function: Full text
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Michael C. Jensen, 1968. "The Performance Of Mutual Funds In The Period 1945–1964," Journal of Finance, American Finance Association, vol. 23(2), pages 389-416, May.
    2. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2012. "Regularization of nonparametric frontier estimators," Journal of Econometrics, Elsevier, vol. 168(2), pages 285-299.
    3. Murty, Sushama & Russell, R. Robert, 2010. "On modeling pollution-generating technologies," The Warwick Economics Research Paper Series (TWERPS) 931, University of Warwick, Department of Economics.
    4. W. Briec & K. Kerstens & J. B. Lesourd, 2004. "Single-Period Markowitz Portfolio Selection, Performance Gauging, and Duality: A Variation on the Luenberger Shortage Function," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 1-27, January.
    5. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2009. "Frontier Estimation and Extreme Values Theory," TSE Working Papers 10-165, Toulouse School of Economics (TSE).
    6. Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2008. "Asymptotics And Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models," Econometric Theory, Cambridge University Press, vol. 24(6), pages 1663-1697, December.
    7. Gregoriou, Greg N. & Sedzro, Komlan & Zhu, Joe, 2005. "Hedge fund performance appraisal using data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 164(2), pages 555-571, July.
    8. Jeong, Seok-Oh & Simar, Léopold, 2006. "Linearly interpolated FDH efficiency score for nonconvex frontiers," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2141-2161, November.
    9. Cinzia Daraio & Léopold Simar, 2005. "Introducing Environmental Variables in Nonparametric Frontier Models: a Probabilistic Approach," Journal of Productivity Analysis, Springer, vol. 24(1), pages 93-121, September.
    10. Badin, Luiza & Daraio, Cinzia & Simar, Léopold, 2010. "Optimal bandwidth selection for conditional efficiency measures: A data-driven approach," European Journal of Operational Research, Elsevier, vol. 201(2), pages 633-640, March.
    11. Wheelock, David C. & Wilson, Paul W., 2008. "Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations," Journal of Econometrics, Elsevier, vol. 145(1-2), pages 209-225, July.
    12. Park, B.U. & Simar, L. & Weiner, Ch., 2000. "The Fdh Estimator For Productivity Efficiency Scores," Econometric Theory, Cambridge University Press, vol. 16(6), pages 855-877, December.
    13. Daraio, Cinzia & Simar, Leopold, 2006. "A robust nonparametric approach to evaluate and explain the performance of mutual funds," European Journal of Operational Research, Elsevier, vol. 175(1), pages 516-542, November.
    14. Léopold Simar & Paul Wilson, 2011. "Inference by the m out of n bootstrap in nonparametric frontier models," Journal of Productivity Analysis, Springer, vol. 36(1), pages 33-53, August.
    15. Jati K. Sengupta, 2000. "Stochastic Efficiency Analysis," World Scientific Book Chapters, in: Dynamic And Stochastic Efficiency Analysis Economics of Data Envelopment Analysis, chapter 4, pages 97-128, World Scientific Publishing Co. Pte. Ltd..
    16. Forsund, Finn R., 2009. "Good Modelling of Bad Outputs: Pollution and Multiple-Output Production," International Review of Environmental and Resource Economics, now publishers, vol. 3(1), pages 1-38, August.
    17. Fare, Rolf, et al, 1993. "Derivation of Shadow Prices for Undesirable Outputs: A Distance Function Approach," The Review of Economics and Statistics, MIT Press, vol. 75(2), pages 374-380, May.
    18. Seok-Oh Jeong & Byeong Park & Léopold Simar, 2010. "Nonparametric conditional efficiency measures: asymptotic properties," Annals of Operations Research, Springer, vol. 173(1), pages 105-122, January.
    19. R. G. Chambers & Y. Chung & R. Färe, 1998. "Profit, Directional Distance Functions, and Nerlovian Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 351-364, August.
    20. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    21. Aragon, Y. & Daouia, A. & Thomas-Agnan, C., 2005. "Nonparametric Frontier Estimation: A Conditional Quantile-Based Approach," Econometric Theory, Cambridge University Press, vol. 21(2), pages 358-389, April.
    22. Fare, Rolf & Grosskopf, Shawna & Noh, Dong-Woon & Weber, William, 2005. "Characteristics of a polluting technology: theory and practice," Journal of Econometrics, Elsevier, vol. 126(2), pages 469-492, June.
    23. Léopold Simar, 2003. "Detecting Outliers in Frontier Models: A Simple Approach," Journal of Productivity Analysis, Springer, vol. 20(3), pages 391-424, November.
    24. Florens, Jean-Pierre & Simar, Leopold, 2005. "Parametric approximations of nonparametric frontiers," Journal of Econometrics, Elsevier, vol. 124(1), pages 91-116, January.
    25. K. Kerstens, 2006. "Multi-horizon Markowitz Portfolio Performance Appraisals," Post-Print hal-00288776, HAL.
    26. Rolf Färe & Shawna Grosskopf, 2000. "Theory and Application of Directional Distance Functions," Journal of Productivity Analysis, Springer, vol. 13(2), pages 93-103, March.
    27. Daouia, Abdelaati & Simar, Leopold, 2007. "Nonparametric efficiency analysis: A multivariate conditional quantile approach," Journal of Econometrics, Elsevier, vol. 140(2), pages 375-400, October.
    28. Cazals, Catherine & Florens, Jean-Pierre & Simar, Leopold, 2002. "Nonparametric frontier estimation: a robust approach," Journal of Econometrics, Elsevier, vol. 106(1), pages 1-25, January.
    29. Morey, Matthew R. & Morey, Richard C., 1999. "Mutual fund performance appraisals: a multi-horizon perspective with endogenous benchmarking," Omega, Elsevier, vol. 27(2), pages 241-258, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Léopold Simar & Paul W. Wilson, 2015. "Statistical Approaches for Non-parametric Frontier Models: A Guided Tour," International Statistical Review, International Statistical Institute, vol. 83(1), pages 77-110, April.
    2. Cinzia Daraio & Léopold Simar & Paul W. Wilson, 2020. "Fast and efficient computation of directional distance estimators," Annals of Operations Research, Springer, vol. 288(2), pages 805-835, May.
    3. Bădin, Luiza & Daraio, Cinzia & Simar, Léopold, 2012. "How to measure the impact of environmental factors in a nonparametric production model," European Journal of Operational Research, Elsevier, vol. 223(3), pages 818-833.
    4. Simar, Léopold & Vanhems, Anne & Wilson, Paul W., 2012. "Statistical inference for DEA estimators of directional distances," European Journal of Operational Research, Elsevier, vol. 220(3), pages 853-864.
    5. Caitlin O’Loughlin & Léopold Simar & Paul W. Wilson, 2023. "Methodologies for assessing government efficiency," Chapters, in: António Afonso & João Tovar Jalles & Ana Venâncio (ed.), Handbook on Public Sector Efficiency, chapter 4, pages 72-101, Edward Elgar Publishing.
    6. Nolwenn Roudaut & Anne Vanhems, 2012. "Explaining firms efficiency in the Ivorian manufacturing sector: a robust nonparametric approach," Journal of Productivity Analysis, Springer, vol. 37(2), pages 155-169, April.
    7. Luiza Bădin & Cinzia Daraio & Léopold Simar, 2014. "Explaining inefficiency in nonparametric production models: the state of the art," Annals of Operations Research, Springer, vol. 214(1), pages 5-30, March.
    8. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2021. "Robustified Expected Maximum Production Frontiers," Econometric Theory, Cambridge University Press, vol. 37(2), pages 346-387, April.
    9. Amir Moradi-Motlagh & Ali Emrouznejad, 2022. "The origins and development of statistical approaches in non-parametric frontier models: a survey of the first two decades of scholarly literature (1998–2020)," Annals of Operations Research, Springer, vol. 318(1), pages 713-741, November.
    10. Amparo Soler Domínguez & Juan Carlos Matallín Sáez & Emili Tortosa Ausina, 2011. "On the informativeness of persistence for mutual funds' performance evaluation using partial frontiers," Working Papers. Serie EC 2011-08, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    11. Frédérique Fève & Jean-Pierre Florens & Léopold Simar, 2023. "Proportional incremental cost probability functions and their frontiers," Empirical Economics, Springer, vol. 64(6), pages 2721-2756, June.
    12. Bădin, Luiza & Daraio, Cinzia & Simar, Léopold, 2019. "A bootstrap approach for bandwidth selection in estimating conditional efficiency measures," European Journal of Operational Research, Elsevier, vol. 277(2), pages 784-797.
    13. Daouia, Abdelaati & Florens, Jean-Pierre & Simar, Léopold, 2020. "Robust frontier estimation from noisy data: A Tikhonov regularization approach," Econometrics and Statistics, Elsevier, vol. 14(C), pages 1-23.
    14. Simar, Léopold & Vanhems, Anne & Van Keilegom, Ingrid, 2016. "Unobserved heterogeneity and endogeneity in nonparametric frontier estimation," Journal of Econometrics, Elsevier, vol. 190(2), pages 360-373.
    15. Daouia, Abdelaati & Simar, Leopold, 2007. "Nonparametric efficiency analysis: A multivariate conditional quantile approach," Journal of Econometrics, Elsevier, vol. 140(2), pages 375-400, October.
    16. Atwood, Joseph & Shaik, Saleem, 2020. "Theory and statistical properties of Quantile Data Envelopment Analysis," European Journal of Operational Research, Elsevier, vol. 286(2), pages 649-661.
    17. Daraio, Cinzia & Simar, Leopold, 2006. "A robust nonparametric approach to evaluate and explain the performance of mutual funds," European Journal of Operational Research, Elsevier, vol. 175(1), pages 516-542, November.
    18. Abdelaati Daouia & Léopold Simar & Paul W. Wilson, 2017. "Measuring firm performance using nonparametric quantile-type distances," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 156-181, March.
    19. Cinzia Daraio & Léopold Simar, 2007. "Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach," Journal of Productivity Analysis, Springer, vol. 28(1), pages 13-32, October.
    20. Diogo Cunha Ferreira & Rui Cunha Marques, 2020. "A step forward on order-α robust nonparametric method: inclusion of weight restrictions, convexity and non-variable returns to scale," Operational Research, Springer, vol. 20(2), pages 1011-1046, June.

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tse:wpaper:23435. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/tsetofr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.