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Computational Statistics

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

Likelihood computation in the normal-gamma stochastic frontier model

  • Bernardo B. de Andrade
  • Geraldo S. Souza
Original Paper
  • 146 Downloads

Abstract

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.

Keywords

Fourier transform Gaussian quadrature Random effects Randomized quasi-Monte Carlo 

Notes

Acknowledgements

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.

References

  1. Abramowitz M, Stegun I (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications, New YorkzbMATHGoogle Scholar
  2. Aigner D, Lovell C, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6(1):21–37MathSciNetCrossRefzbMATHGoogle Scholar
  3. Beckers DE, Hammond CJ (1987) A tractable likelihood function for the normal-gamma stochastic frontier model. Econ Lett 24(1):33–38MathSciNetCrossRefzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle 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, BerlinzbMATHGoogle Scholar
  8. Greene WH (1990) A gamma-distributed stochastic frontier model. J Econom 46(1–2):141–163MathSciNetCrossRefzbMATHGoogle 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. http://www.limdep.com/
  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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Krommer A, Ueberhuber C (1998) Computational integration. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  14. Kumbhakar S, Lovell C (2003) Stochastic frontier analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  15. Laurie DP (1997) Calculation of Gauss–Kronrod quadrature rules. Math Comput Am Math Soc 66:1133–1145MathSciNetCrossRefzbMATHGoogle 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 CrossRefzbMATHGoogle Scholar
  17. Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb–Douglas production functions with composed error. Int Econ Rev 18(2):435–444CrossRefzbMATHGoogle 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 CrossRefzbMATHGoogle Scholar
  19. Monahan JF (2011) Numerical methods of statistics, 2nd edn. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  20. Piessens R, De Doncker-Kapenga E, Uberhuber C (1983) Quadpack: a subroutine package for automatic integration. Springer, BerlinCrossRefzbMATHGoogle 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/j.cam.2006.11.021 MathSciNetCrossRefzbMATHGoogle 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 23:661–671MathSciNetzbMATHGoogle Scholar
  24. Stevenson RE (1980) Likelihood functions for generalized stochastic frontier estimation. J Econom 13(1):57–66CrossRefzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Tuffin B (2004) Randomization of quasi-Monte Carlo methods for error estimation: survey and normal approximation. Monte Carlo Methods Appl 3(4):617–628MathSciNetzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle 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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