The Distribution of the Concentration Ratio for Samples from a Uniform Population

Giovanni Girone, Antonella Nannavecchia

Faculty of Economics, University of Bari, Bari, Italy.

DOI: 10.4236/am.2015.61007                

Abstract

In the present paper we derived, with direct method, the exact expressions for the sampling probability density function of the Gini concentration ratio for samples from a uniform population of size n = 6, 7, 8, 9 and 10. Moreover, we found some regularities of such distributions valid for any sample size.

Keywords

Gini Concentration Ratio, Uniform Distribution, Order Statistics, Probability Density Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Gini, C. (1914) L’ammontare e la composizionedellaricchezzadellenazioni. Bocca, Torino.
[2]	Hoeffding, W. (1948) A Class of Statistics with Asymptotically Normal Distribution. Annals of Mathematical Statistics, 19, 293-325.
[3]	Glasser, G.J. (1962) Variance Formulas for the Mean Difference and the Coefficient of Concentration. Journal of the American Statistical Association, 57, 648-654. http://dx.doi.org/10.1080/01621459.1962.10500553
[4]	Cucconi, O. (1965) Sulla distribuzionecampionaria del rapporto R di concentrazione. Statistica, 25, 119.
[5]	Dall’Aglio, G. (1965) Comportamentoasintoticodellestimedelladifferenza media e del rapporto di concentrazione. Metron, 24, 379-414.
[6]	Deltas, G. (2003) The Small-Sample Bias of the Gini Coefficient: Results and Implications for Empirical Research. Review of Economics and Statistics, 85, 226-234. http://dx.doi.org/10.1162/rest.2003.85.1.226
[7]	Barrett, G.F. and Donald, S.G. (2009) Statistical Inference with Generalized Gini Indices of Inequality, Poverty, and Welfare. Journal of Business & Economic Statistics, 27, 1-17. http://dx.doi.org/10.1198/jbes.2009.0001
[8]	Davidson, R. (2009) Reliable Inference for the Gini Index. Journal of Econometrics, 150, 30-40. http://dx.doi.org/10.1016/j.jeconom.2008.11.004
[9]	Fakoor, V., Ghalibaf, M.B. and Azarnoosh, H.A. (2011) Asymptotic Behaviors of the Lorenz Curve and Gini Index in Sampling from a Length-Biased Distribution. Statistics and Probability Letters, 81, 1425-1435. http://dx.doi.org/10.1016/j.spl.2011.04.013
[10]	Girone, G. (1968) Sulcomportamentocampionariosimulato del rapporto di concentrazione. Annalidella Facoltà di Economia e Commerciodell’Universitàdegli Studi di Bari, 23, 5-11.
[11]	Girone, G. (1971) La distribuzione del rapporto di concentrazione per campionicasuali di variabiliesponenziali. Studi di Probabilità, Statistica e Ricercaoperativa in onore di Giuseppe Pompilj, Oderisi, Gubbio.
[12]	Girone, G. (1971) La distribuzione del rapporto di concentrazione per piccolicampioniestratti da unapopolazioneuniforme. Annalidell’Istituto di Statisticadell’Universitàdegli Studi di Bari, 36, 31-52.