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GMM Estimation of Mixtures from Grouped Data:

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  • William E. Griffiths and Gholamreza Hajargasht

Abstract

We show how the generalized method of moments (GMM) framework developed in Hajargasht et al. (2012) for estimating income distributions from grouped data can be adapted for estimating mixtures. This approach can be used to estimate a mixture of any distributions where the moments and moment distribution functions of the mixture components can be expressed in terms of the parameters of those components. The required expressions for mixtures of lognormal and gamma densities are provided; in our empirical work we focus on estimation of mixtures of lognormal distributions. Two- and three-component lognormal mixtures are estimated for the income distributions of China rural, China urban, India rural, India urban, Pakistan, Russia, South Africa, Brazil and Indonesia. Their performance, in terms of goodness-of-fit and validity of moment conditions, is compared with that of a generalized beta (GB2) distribution. We find that the three-component lognormal mixture always outperforms the GB2 distribution, but the two-component mixture does not. For Brazil and Indonesia we have single observations, making it possible to compare maximum likelihood estimation of the mixtures from a complete set of single observations with GMM estimates obtained after grouping the data. Estimates from both procedures are found to be comparable, lending support to the usefulness of the GMM approach.

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File URL: http://www.economics.unimelb.edu.au/research/1148.pdf
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Bibliographic Info

Paper provided by The University of Melbourne in its series Department of Economics - Working Papers Series with number 1148.

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Length: 32 pages
Date of creation: 2012
Date of revision:
Handle: RePEc:mlb:wpaper:1148

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Related research

Keywords: Lognormal distribution; Generalized beta distribution; Inequality measures;

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References

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  1. Gholamreza Hajargasht & William E. Griffiths & Joseph Brice & D.S. Prasada Rao & Duangkamon Chotikapanich, 2012. "Inference for Income Distributions Using Grouped Data," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(4), pages 563-575, May.
  2. Russell Davidson, 2007. "Reliable Inference For The Gini Index," Working Papers halshs-00353856, HAL.
  3. Duangkamon Chotikapanich & William E. Griffiths & D. S. Prasada Rao & Vicar Valencia, 2012. "Global Income Distributions and Inequality, 1993 and 2000: Incorporating Country-Level Inequality Modeled with Beta Distributions," The Review of Economics and Statistics, MIT Press, vol. 94(1), pages 52-73, February.
  4. Chotikapanich, Duangkamon & Griffiths, William E. & Rao, D. S. Prasada, 2007. "Estimating and Combining National Income Distributions Using Limited Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 25, pages 97-109, January.
  5. Michel Lubrano & Abdoul Aziz Junior Ndoye, 2011. "Inequality decomposition using the Gibbs output of a Mixture of lognormal distributions," Working Papers halshs-00585248, HAL.
  6. repec:hal:cesptp:halshs-00175923 is not listed on IDEAS
  7. Hasegawa, Hikaru & Kozumi, Hideo, 2003. "Estimation of Lorenz curves: a Bayesian nonparametric approach," Journal of Econometrics, Elsevier, vol. 115(2), pages 277-291, August.
  8. Flachaire, Emmanuel & Nunez, Olivier, 2007. "Estimation of the income distribution and detection of subpopulations: An explanatory model," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3368-3380, April.
  9. Duangkamon Chotikapanich & William E Griffiths, 2008. "Estimating Income Distributions Using a Mixture of Gamma Densities," Department of Economics - Working Papers Series 1034, The University of Melbourne.
  10. Emmanuel Flachaire & Olivier G. Núñez, 2003. "Estimation Of Income Distribution And Detection Of Subpopulations: An Explanatory Model," Statistics and Econometrics Working Papers ws030201, Universidad Carlos III, Departamento de Estadística y Econometría.
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