# Bayesian Estimation of Beta-type Distribution Parameters Based on Grouped Data

## Author

Listed:
• Kazuhiko Kakamu

(Kobe University)

• Haruhisa Nishino

(Hiroshima University)

## Abstract

This study considers a method of estimating generalized beta (GB) distribution parameters based on grouped data from a Bayesian point of view and explores the possibility of the GB distribution focusing on the goodness-of-fit because the GB distribution is one of the most typical five-parameter distributions. It uses a tailored randomized block Metropolis–Hastings (TaRBMH) algorithm to estimate the GB distribution parameters and this method is then applied to one simulated and two real datasets. Moreover, the fit of the GB distribution is compared with those of the generalized beta distribution of the second kind (GB2 distribution) and Dagum (DA) distribution by using the marginal likelihood. The estimation results of simulated and real datasets show that the GB distributions are preferred to the DA distributions in general, while the GB2 distributions have similar performances to the GB distributions. In other words, the GB2 distribution could be adopted as well as the GB distribution in terms of the smallest possible number of parameters, although our TaRBMH algorithm can estimate the GB distribution parameters efficiently and accurately. The accuracy of the Gini coefficients also suggests the use of the GB2 distributions.

## Suggested Citation

• Kazuhiko Kakamu & Haruhisa Nishino, 2019. "Bayesian Estimation of Beta-type Distribution Parameters Based on Grouped Data," Computational Economics, Springer;Society for Computational Economics, vol. 54(2), pages 625-645, August.
• Handle: RePEc:kap:compec:v:54:y:2019:i:2:d:10.1007_s10614-018-9843-4
DOI: 10.1007/s10614-018-9843-4
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## References listed on IDEAS

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1. Wichitaksorn, Nuttanan & Tsurumi, Hiroki, 2013. "Comparison of MCMC algorithms for the estimation of Tobit model with non-normal error: The case of asymmetric Laplace distribution," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 226-235.
2. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
3. Parker, Simon C, 1999. "The Beta as a Model for the Distribution of Earnings," Bulletin of Economic Research, Wiley Blackwell, vol. 51(3), pages 243-251, July.
4. Christian Kleiber, 2008. "A Guide to the Dagum Distributions," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 6, pages 97-117, Springer.
5. Majumder, Amita & Chakravarty, Satya Ranjan, 1990. "Distribution of Personal Income: Development of a New Model and Its Application to U.S. Income Data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 5(2), pages 189-196, April-Jun.
6. Salem, A B Z & Mount, T D, 1974. "A Convenient Descriptive Model of Income Distribution: The Gamma Density," Econometrica, Econometric Society, vol. 42(6), pages 1115-1127, November.
7. F. Clementi & M. Gallegati & G. Kaniadakis & S. Landini, 2016. "$\kappa$-generalized models of income and wealth distributions: A survey," Papers 1610.08676, arXiv.org.
8. Kloek, Teun & van Dijk, Herman K., 1978. "Efficient estimation of income distribution parameters," Journal of Econometrics, Elsevier, vol. 8(1), pages 61-74, August.
9. Goffe, William L. & Ferrier, Gary D. & Rogers, John, 1994. "Global optimization of statistical functions with simulated annealing," Journal of Econometrics, Elsevier, vol. 60(1-2), pages 65-99.
10. Hasegawa, Hikaru & Kozumi, Hideo, 2003. "Estimation of Lorenz curves: a Bayesian nonparametric approach," Journal of Econometrics, Elsevier, vol. 115(2), pages 277-291, August.
11. Chib, Siddhartha & Ramamurthy, Srikanth, 2010. "Tailored randomized block MCMC methods with application to DSGE models," Journal of Econometrics, Elsevier, vol. 155(1), pages 19-38, March.
12. Duangkamon Chotikapanich & D. S. Prasada Rao & Kam Ki Tang, 2007. "Estimating Income Inequality In China Using Grouped Data And The Generalized Beta Distribution," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 53(1), pages 127-147, March.
13. Gastwirth, Joseph L, 1972. "The Estimation of the Lorenz Curve and Gini Index," The Review of Economics and Statistics, MIT Press, vol. 54(3), pages 306-316, August.
14. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 69(2), pages 427-428, October.
15. van Dijk, Herman K & Kloek, Teun, 1980. "Inferential Procedures in Stable Distributions for Class Frequency Data on Incomes," Econometrica, Econometric Society, vol. 48(5), pages 1139-1148, July.
16. Dorfman, Robert, 1979. "A Formula for the Gini Coefficient," The Review of Economics and Statistics, MIT Press, vol. 61(1), pages 146-149, February.
17. McDonald, James B & Mantrala, Anand, 1995. "The Distribution of Personal Income: Revisited," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 10(2), pages 201-204, April-Jun.
18. Singh, S K & Maddala, G S, 1976. "A Function for Size Distribution of Incomes," Econometrica, Econometric Society, vol. 44(5), pages 963-970, September.
19. McDonald, James B & Ransom, Michael R, 1979. "Functional Forms, Estimation Techniques and the Distribution of Income," Econometrica, Econometric Society, vol. 47(6), pages 1513-1525, November.
20. Kazuhiko Kakamu, 2016. "Simulation Studies Comparing Dagum and Singh–Maddala Income Distributions," Computational Economics, Springer;Society for Computational Economics, vol. 48(4), pages 593-605, December.
21. Parker, Simon C., 1999. "The generalised beta as a model for the distribution of earnings," Economics Letters, Elsevier, vol. 62(2), pages 197-200, February.
22. Robert F. Bordley & James B. McDonald & Anand Mantrala, 1997. "Something New, Something Old: Parametric Models for the Size of Distribution of Income," Journal of Income Distribution, Ad libros publications inc., vol. 6(1), pages 5-5, June.
23. 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.
24. Slottje, Daniel J., 1984. "A measure of income inequality in the U.S. for the years 1952-1980 based on the beta distribution of the second kind," Economics Letters, Elsevier, vol. 15(3-4), pages 369-375.
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