Computational Statistics

, Volume 33, Issue 2, pp 967–982 | Cite as

Likelihood computation in the normal-gamma stochastic frontier model

Original Paper


Likelihood-based estimation of the normal-gamma stochastic frontier model requires numerical integration to solve its likelihood. For the integration methods found in the literature, it is not known under which conditions they perform optimally or if there is a method that performs better than the others. Our aim is to study the applicability of available methods and to compare them based on their ability to approximate the loglikelihood. We consider three principles—numerical quadrature, inversion of the characteristic function and Monte Carlo—and assess the effect of the parameters on the accuracy of each of six numerical procedures.


Fourier transform Gaussian quadrature Random effects Randomized quasi-Monte Carlo 



GSS is grateful to the National Council for Scientific and Technological Development—CNPq/Brazil, for financial support. BBA has been partially funded by the Federal District Research Foundation, FAP/DF.


  1. Abramowitz M, Stegun I (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications, New YorkMATHGoogle Scholar
  2. Aigner D, Lovell C, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6(1):21–37MathSciNetCrossRefMATHGoogle Scholar
  3. Beckers DE, Hammond CJ (1987) A tractable likelihood function for the normal-gamma stochastic frontier model. Econ Lett 24(1):33–38MathSciNetCrossRefMATHGoogle Scholar
  4. Cohen AC (1950) Estimating the mean and variance of normal populations from singly truncated and doubly truncated samples. Ann Math Stat 21(4):557–569. doi: 10.1214/aoms/1177729751 MathSciNetCrossRefMATHGoogle Scholar
  5. Cullinane K, Wang TF, Song DW, Ji P (2006) The technical efficiency of container ports: comparing data envelopment analysis and stochastic frontier analysis. Transp Res Part A Policy Pract 40(4):354–374. doi: 10.1016/j.tra.2005.07.003 CrossRefGoogle Scholar
  6. Fried H, Lovell C, Schmidt S (2008) The measurement of productive efficiency and productivity growth. Oxford University Press, OxfordCrossRefGoogle Scholar
  7. Glasserman P (2004) Monte Carlo methods in financial engineering. Springer, BerlinMATHGoogle Scholar
  8. Greene WH (1990) A gamma-distributed stochastic frontier model. J Econom 46(1–2):141–163MathSciNetCrossRefMATHGoogle Scholar
  9. Greene WH (2003) Simulated likelihood estimation of the normal-gamma stochastic frontier function. J Prod Anal 19(2):179–190. doi: 10.1023/A:1022853416499 CrossRefGoogle Scholar
  10. Greene WH (2012) LIMDEP 10, Econometric Software, Inc.
  11. Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19(2–3):233–238MathSciNetCrossRefGoogle Scholar
  12. Kozumi H, Zhang X (2005) Bayesian and non-bayesian analysis of gamma stochastic frontier models by Markov Chain Monte Carlo methods. Comput Stat 20(4):575–593. doi: 10.1007/BF02741316 MathSciNetCrossRefMATHGoogle Scholar
  13. Krommer A, Ueberhuber C (1998) Computational integration. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  14. Kumbhakar S, Lovell C (2003) Stochastic frontier analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  15. Laurie DP (1997) Calculation of Gauss–Kronrod quadrature rules. Math Comput Am Math Soc 66:1133–1145MathSciNetCrossRefMATHGoogle Scholar
  16. Matsumoto M, Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans Model Comput Simul 8(1):3–30. doi: 10.1145/272991.272995 CrossRefMATHGoogle Scholar
  17. Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb–Douglas production functions with composed error. Int Econ Rev 18(2):435–444CrossRefMATHGoogle Scholar
  18. Mittnik S, Doganoglu T, Chenyao D (1999) Computing the probability density function of the stable paretian distribution. Math Comput Model 29(10):235–240. doi: 10.1016/S0895-7177(99)00106-5 CrossRefMATHGoogle Scholar
  19. Monahan JF (2011) Numerical methods of statistics, 2nd edn. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  20. Piessens R, De Doncker-Kapenga E, Uberhuber C (1983) Quadpack: a subroutine package for automatic integration. Springer, BerlinCrossRefMATHGoogle Scholar
  21. Ritter C, Simar L (1997) Pitfalls of normal-gamma stochastic frontier models. J Prod Anal 8(2):167–182. doi: 10.1023/A:1007751524050 CrossRefGoogle Scholar
  22. Shampine L (2008) Vectorized adaptive quadrature in MATLAB. J Comput Appl Math 211(2):131–140. doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  23. Steen NM, Byrne GD, Gelbard EM (1969) Gaussian quadratures for the integrals \(\int _{0}^{\infty }\,\text{ exp }(-x^{2})f(x)dx\) and \( \int _{0}^{b}\,\text{ exp }(-x^{2})f(x)dx\). Math Comput 23:661–671MathSciNetMATHGoogle Scholar
  24. Stevenson RE (1980) Likelihood functions for generalized stochastic frontier estimation. J Econom 13(1):57–66CrossRefMATHGoogle Scholar
  25. Tsionas EG (2012) Maximum likelihood estimation of stochastic frontier models by the Fourier transform. J Econom 170(1):234–248. doi: 10.1016/j.jeconom.2012.04 MathSciNetCrossRefMATHGoogle Scholar
  26. Tuffin B (2004) Randomization of quasi-Monte Carlo methods for error estimation: survey and normal approximation. Monte Carlo Methods Appl 3(4):617–628MathSciNetMATHGoogle Scholar
  27. Warr RL (2014) Numerical approximation of probability mass functions via the inverse discrete Fourier transform. Methodol Comput Appl Probab 16(4):1025–1038. doi: 10.1007/s11009-013-9366-3 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of BrasíliaBrasíliaBrazil
  2. 2.Brazilian Agricultural Research Corporation (EMBRAPA)Parque Estação Biológica, Asa NorteBrasíliaBrazil

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