Abstract
In this paper we deal with the distribution of the Gini Concentration Index (G). We derive its exact sample distribution which is useful only for small sample sizes and basing on the classic results for L-statistics of Egorov and Nevronov we prove the asymptotic normality for G from an original and probabilistic way.
References
Crocetta, C.; Loperfido, N. (2005) “The exact sampling distribution of L-statistics”, Metron 63(2): 213–223.
Egorov, V. A. & Nevzorov, V. B. (1976) Limit theorems for linear combinations of order statistics. Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), pp. 63–79. Lecture Notes in Math., Vol. 550, Springer, Berlin, 1976, 7: 143–151.
Giles, E.A.D. (2004) “Calculating a standard error for the Gini coefficient: some further results” Oxford Bulletin of Economics & Statistics 66(3): 425–428.
Gini, C. (1995) Variabilitá e mutabilitá. 1912. Reprinted in: E. Pizetti and T. Salvemini (Eds.) Memorie di Metodoloǵıa Statistica, Libreŕıa Eredi Virgilio Veschi, Rome.
Glasser, G.J. (1962) Variance Formulas for the Mean Difference and Coefficient of Concentration. J. Amer. Stat. Assoc. 57(299): 648–654.
Lerman, R.I.; Yitzhaki, S. (1984) “A note on the calculation and interpretation of the Gini index”, Economics Letters 5: 363–368.
Martínez-Camblor, P. (2005) Normalidad asintótica para los E-Gini. Revista de la Sociedad Argentina de Estadística 9: 35–42.
Martínez-Camblor, P. (2006) Comparando ı́ndices de desigualdad de Theil en muestras relacionadas, Revista de la Sociedad Argentina de Estad́ıstica, en prensa.
Martínez-Camblor, P. (2007) Central Limit Theorems for S-Gini and Theil Coefficients. Revista Colombiana de Estadística, 30(2): 287–300.
Martínez-Camblor, P. (2007) “Comparing S-Gini and E-Gini on Paired Samples”, InterStat Journal, May, 11 pages. http://interstat.statjournals.net/YEAR/2007/articles/0705001.pdf
Sen, A. (1973) On Economic Inequality. Clarendon Press, Oxford.
Vandemaele, M.; Veraverbeke, N. (1982) “Cramér type large deviations for linear combinations of order statistics”, Annals of Probability 10(2): 423–434.
Zitikis, R.; Gastwirth, J.L. (2002) “Asymptotic distribution of the S-Gini index”, Australian & New Zealand Journal of Statistics 44(4): 439–446.
Zitikis, R. (2003) “Asymptotic estimation of the E-Gini index”, Econometric Theory 19: 587–601.
Comments
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Copyright (c) 2009 Revista de Matemática: Teoría y Aplicaciones