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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.

Suggested Citation

  • William E. Griffiths and Gholamreza Hajargasht, 2012. "GMM Estimation of Mixtures from Grouped Data:," Department of Economics - Working Papers Series 1148, The University of Melbourne.
    • Handle: RePEc:mlb:wpaper:1148
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    1. Branko Milanovic, 2002. "True World Income Distribution, 1988 and 1993: First Calculation Based on Household Surveys Alone," Economic Journal, Royal Economic Society, vol. 112(476), pages 51-92, January.
    2. 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.
    3. 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.
    4. Duangkamon Chotikapanich & William E. Griffiths, 2008. "Estimating Income Distributions Using a Mixture of Gamma Densities," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 16, pages 285-302, Springer.
    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. Davidson, Russell, 2009. "Reliable inference for the Gini index," Journal of Econometrics, Elsevier, vol. 150(1), pages 30-40, May.
    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. 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.
    9. Alex Bakker & John Creedy, 1999. "Macroeconomic variables and income inequality in New Zealand: An exploration using conditional mixture distributions," New Zealand Economic Papers, Taylor & Francis Journals, vol. 33(2), pages 59-79.
    10. 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.
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    Cited by:

    1. Kazuhiko Kakamu, 2022. "Bayesian analysis of mixtures of lognormal distribution with an unknown number of components from grouped data," Papers 2210.05115, arXiv.org, revised Sep 2023.
    2. Tobias Eckernkemper & Bastian Gribisch, 2021. "Classical and Bayesian Inference for Income Distributions using Grouped Data," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 83(1), pages 32-65, February.
    3. Duangkamon Chotikapanich & William E. Griffiths & Gholamreza Hajargasht & D. S. Prasada Rao & Charley Xia, 2018. "Inequality and Poverty in Africa: Comparing Panels of Income Distributions from Different Data Sources," Department of Economics - Working Papers Series 2042, The University of Melbourne.

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    Keywords

    Lognormal distribution; Generalized beta distribution; Inequality measures;
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