Abstract
In this paper we estimate income distributions, Lorenz curves and the related Gini index using a Bayesian nonparametric approach based on Polya tree priors. In particular, we propose an alternative approach for dealing with contaminated observations and extreme income values: avoiding the common practise that removes these critical data, we instead treat them as censored observations and apply a Polya tree model for incomplete data. The proposed method is illustrated through an empirical application based on the European Survey on Income Living Conditions data.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We thank an anonymous referee for mentioning this important point.
We thank an anonymous referee for this point.
References
Blum, J., Susarla, V.: On the posterior distribution of a Dirichlet process given randomly right censored observations. Stoch. Process. Appl. 5, 207–211 (1977)
Burkhauser, R.V., Feng, S., Jenkins, S., Larrimore, J.: Estimating trends in US income inequality using the Current Population Survey: the importance of controlling for censoring. J. Econ. Inequal. 9, 393–415 (2011)
Chotikapanich, D., Griffiths, W.: Estimating Lorenz curves using Dirichlet distribution. J. Bus. Econ. Stat. 20, 290–295 (2002)
Cowell, F., Flaichaire, E.: Income distribution and inequality measurement: the problem of extreme values. J. Econom. 141, 1044–1072 (2007)
Cowell, F., Victoria Feser, M.P.: Statistical inference for Lorenz curves with censored data. Discussion Paper No. DARP/35 (1998)
Cowell, F., Victoria-Feser, M.P.: Distributional dominance with trimmed data. J. Bus. Econ. Stat. 24, 291–300 (2006)
De Finetti, B.: Il problema della perequazione. In: Atti della Società Italiana per il Progresso delle Scienze, (XII Riunione), Napoli (1935)
Doksum, K.A.: Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2, 183–201 (1974)
Fabius, J.: Asymptotic behavior of Bayes estimate. Ann. Math. Stat. 35, 846–856 (1964)
Ferguson, T.S.: A Bayesian analysis on some nonparametric problems. Ann. Stat. 1, 209–230 (1973)
Ferguson, T.S.: Prior distributions on spaces of probability measures. Ann. Stat. 2, 615–629 (1974)
Fienberg, S.E., Steele, R.J., Makov, U.E.: Statistical notion of data disclosure avoidance and their relationship to traditional statistical methodology: data swapping and loglinear models. In: Proceedings of the Bureau of the Census 12th Annual Research Conference, pp. 87–105 (1996)
Freedman, D.A.: On the asymptotic behavior of Bayes’ estimate in the discrete case. Ann. Math. Stat. 34, 1386–1403 (1963)
Gastwirth, J.L.: A general definition of the Lorenz curve. Econometrica 31, 1037–1039 (1971)
George, E.I., Hui, S.K.: Optimal pricing using online auction experiments: a Polya tree approach. Ann. Appl. Stat. 6, 55–82 (2012)
Ghosh, J.K., Ramamoorthi, R.V.: Bayesian Nonparametrics. Springer, New York (2003)
Gini, C.: Variabilità e mutabilità. In: Contributo allo studio delle distribuzioni e relazioni statistiche. Studi Economico-Giuridici dell’Università di Cagliari III (1912)
Gini, C.: Sulla misura della concentrazione e della Variabilità dei caratteri. In: Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti LXXIII(part 2), pp. 1203–1248 (1914)
Gottschak, P., Smeeding, T.M.: Empirical evidence in income inequality in industrialized countries. In: Atkinson, A., Bourguignon, F. (eds.) Handbook of Income Distribution, Chap. 3. North-Holland, Amsterdam
Hanson, T., Johnson, W.O.: Modeling regression error with a mixture of Polya trees. J. Am. Stat. Assoc. 101, 1548–1565 (2002)
Hasegawa, H., Kozumi, H.: Estimation of Lorenz curves: a Bayesian nonparametric approach. J. Econ. 115, 277–291 (2003)
Kendall, M., Stuart, A.: The Advanced Theory of Statistics. Mac Millan Publishing, New York (1977)
Lavine, M.: Some aspects of Polya tree distributions for statistical modelling. Ann. Stat. 20, 1222–1235 (1992)
Lavine, M.: More aspects of Polya tree distributions for statistical modelling. Ann. Stat. 22, 1161–1176 (1994)
Lorenz, M.O.: Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc. 9, 209–219 (1905)
Mauldin, D., Sudderth, W.D., Williams, S.C.: Polya trees and random distributions. Ann. Stat. 20, 1203–1221 (1992)
Müller, P., Quintana, F.: More nonparametric Bayesian models for biostatistics. In: Hjort, N.L., Holmes, C., Müller, P., Walker, S.G. (eds.) Bayesian Nonparametrics. Cambridge University Press, Cambridge (2010)
Muliere, P., Walker, S.: A Bayesian non-parametric approach to survival analysis using Polya trees. Scand. J. Stat. 24, 331–340 (1997)
Neath, A.A.: Polya tree distributions for statistical modeling of censored data. J. Appl. Math. Decis. Sci. 7, 175–186 (2003)
Nieto-Barajas, L.E., Müller, P.: Rubbery Polya tree. Scand. J. Stat. 39, 166–184 (2012)
Pietra, G.: Delle relazioni tra gli indici di Variabilità, I, pp. 775–804. II. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti LXXIV(II) (1915)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, New York (2004)
Susarla, V., Van Ryzin, J.: Non parametric Bayesian estimation of survival curves from incomplete observations. J. Am. Stat. Assoc. 71, 897–902 (1976)
Van Kerm, P.: Extreme incomes and the estimation of poverty and inequality indicators from EU-SILC. IRISS Working Paper Series 2007–01, CEPS/INSTEAD, Differdange, Luxembourg (2007)
Walker, S., Mallick, B.: Hierarchical generalized linear models and frailty models with Bayesian nonparametric mixing. J. R. Stat. Soc. Ser. B 59, 845–860 (1997)
Acknowledgments
We are very grateful to two anonymous referees for much appreciated comments and advices.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gigliarano, C., Muliere, P. Estimating the Lorenz curve and Gini index with right censored data: a Polya tree approach. METRON 71, 105–122 (2013). https://doi.org/10.1007/s40300-013-0009-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40300-013-0009-9