Journal of Productivity Analysis

, Volume 43, Issue 2, pp 215–223 | Cite as

Closed-form solution for a bivariate distribution in stochastic frontier models with dependent errors

  • Emilio Gómez-Déniz
  • Jorge V. Pérez-Rodríguez


This paper proposes a bivariate continuous model based on normal–half normal distributions for testing the independence of idiosyncratic and inefficiency terms in the stochastic frontier model in a maximum likelihood framework. This model allows us to construct a closed-form of the marginal distribution of the composite error term dependent on a parameter which gives a flexible covariance structure (positive and negative correlations are possible), but also nests classical models utilised in stochastic frontier studies. In addition, we obtain the point estimator for technical efficiency using the Battese and Coelli (J Econom 38:387–399, 1988) expression.


Technical and cost efficiencies Stochastic frontier Marginal distribution Dependence Sarmanov model 

JEL Classification

C01 C13 C21 C51 



The research was partially funded by Emilio Gómez-Déniz: ECO2009-14152 (MICINN, Spain) and J.V. Pérez-Rodríguez: ECO2011-23189 (Ministerio de Economía y Competitividad, Spain). Also, J.V. Pérez-Rodríguez acknowledges the Department of Statistics and CRiSM, at the University of Warwick for their special support, since part of this paper was written while he was visiting the University of Warwick in 2013.


  1. Aigner D, Lovell K, Schmidt P (1977) Formulation and estimation of stochastic frontier function models. J Econom 6:21–37CrossRefGoogle Scholar
  2. Ali MM, Mikhail NN, Haq MS (1978) A class of bivariate distributions including the bivariate logistic. J Multivar Anal 8:405–412CrossRefGoogle Scholar
  3. Amblard C, Girard S (2009) A new extension of bivariate FGM copulas. Metrika 70:1–17CrossRefGoogle Scholar
  4. Arnold BC, Castillo E, Sarabia JM (1999) Conditional specification of statistical models. Springer series in statistics. Springer, New YorkGoogle Scholar
  5. Arnold BC, Castillo E, Sarabia JM (2001) Conditionally specified distributions: an introduction (with discussion). Stat Sci 16(3):249–274CrossRefGoogle Scholar
  6. Bairamov I, Kotz S, Gebizlioglu OL (2000) The Sarmanov family and its generalization. S Afr Stat J 35:205–224Google Scholar
  7. Bairamov I, Altinsoy B, Kerns G (2011) On generalized Sarmanov bivariate distributions. TWMS J Appl Eng Math 1(1):86–97Google Scholar
  8. Balakrishnan N, Lai C-D (2009) Continuous bivariate distributions, 2nd edn. Springer, BerlinGoogle Scholar
  9. Battese G, Corra G (1977) Estimation of a production frontier model: with application to the pastoral zone of Eastern Australia. Aust J Agric Econ 21(3):169–179CrossRefGoogle Scholar
  10. Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econom 38:387–399CrossRefGoogle Scholar
  11. Cameron AC, Trivedi PK (1993) Tests of independence in parametric models with applications and illustrations. J Bus Econ Stat 11(1):29–43Google Scholar
  12. Cameron AC, Trivedi PK (1998) Regression analysis of count data. Cambridge University Press, New YorkCrossRefGoogle Scholar
  13. Cohen L (1984) Probability distributions with given multivariate marginals. J Math Phys 25:2402–2403CrossRefGoogle Scholar
  14. Cox DR, Hinkley DV (1974) Theor Stat. Chapman and Hall, LondonCrossRefGoogle Scholar
  15. Danaher PJ, Hardie BGS (2005) Bacon with your eggs? Applications of a new bivariate beta-binomial distribution. Am Stat 59(4):282–286CrossRefGoogle Scholar
  16. Danaher PJ, Smith PS (2011) Modeling multivariate distributions using copulas: applications in marketing. Mark Sci 30:4–21CrossRefGoogle Scholar
  17. Drouet D, Kotz S (2001) Correlation and dependence. Imperial College Press, SingaporeGoogle Scholar
  18. Eagleson GK (1964) Polynomial expansions of bivariate distributions. Ann Math Stat 35:1208–1215CrossRefGoogle Scholar
  19. Frank MJ (1979) On the simultaneous associativity of \(F(x, y)\) and \(x+y-F(x, y)\). Aequationes Mathematicae 19:194–226CrossRefGoogle Scholar
  20. Greene W (1980a) Maximum likelihood estimation of econometric frontier functions. J Econom 13(1):27–56CrossRefGoogle Scholar
  21. Greene W (1980b) On the estimation of a flexible frontier production model. J Econom 13(1):101–115CrossRefGoogle Scholar
  22. Greene W (1990) A gamma distributed stochastic frontier model. J Econom 46(1):141–164CrossRefGoogle Scholar
  23. Greene W (2003) Maximum simulated likelihood estimation of the normal-gamma stochastic frontier function. J Prod Anal 19(2–3):179–190CrossRefGoogle Scholar
  24. Gupta PL, Gupta RC, Tripathi RC (2004) Score test for zero inflated generalized Poisson regression model. Commun Stat Theory Methods 33(1):47–64CrossRefGoogle Scholar
  25. Hernández A, Fernández MP, Gómez-Déniz E (2009) The net Bayes premium with dependence between the risk profiles. Insur Math Econ 45:247–254CrossRefGoogle Scholar
  26. Kumbhakar SC, Lovell CA (2000) Stochastic frontiers analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  27. Lancaster HO (1958) The structure of bivariate distributions. Ann Math Stat 29:719–736CrossRefGoogle Scholar
  28. Lee L-F (1983) A test for distributional assumptions for the stochastic frontier functions. J Econom 22(3):245–267CrossRefGoogle Scholar
  29. Lee TML (1996) Properties and applications of the Sarmanov family of bivariate distributions. Commun Stat Theory Methods 25(6):1207–1222CrossRefGoogle Scholar
  30. Meeusen W, Van Den Broeck J (1977) Efficiency estimation from Cobb-Douglas production function with composed error. Int Econ Rev 18:435–444CrossRefGoogle Scholar
  31. Nelsen RB (1998) An introduction to Copulas. Springer, New YorkGoogle Scholar
  32. Park Y-H, Fader PS (2004) Modeling browsing behavior at multiple websites. Mark Sci 23(3):280–303CrossRefGoogle Scholar
  33. Sarabia JM, Gómez-Déniz E (2011) Multivariate Poisson-Beta distributions with applications. Commun Stat Theory Methods 40:1093–1108CrossRefGoogle Scholar
  34. Sarmanov OV (1966) Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168:596–599Google Scholar
  35. Shubina Maria, Lee TML (2004) On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions. Commun Stat Theory Methods 33(5):1031–1052CrossRefGoogle Scholar
  36. Smith M (2008) Stochastic frontier models with dependent error components. Econom J 11:172–192CrossRefGoogle Scholar
  37. Stevenson R (1980) Likelihood functions for generalized stochastic frontier functions. J Econom 13:57–66CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Emilio Gómez-Déniz
    • 1
  • Jorge V. Pérez-Rodríguez
    • 2
  1. 1.Department of Quantitative Methods in Economics and TiDES InstituteUniversity of Las Palmas de Gran CanariaLas Palmas de Gran CanariaSpain
  2. 2.Department of Quantitative Methods in EconomicsUniversity of Las Palmas de Gran CanariaLas Palmas de Gran CanariaSpain

Personalised recommendations