1Fundación Caubet-Cimera Illes Balears, Mallorca, España. Programa de epidemiología e investigación clínica. Email: martinez@caubet-cimera.es
]]> Abstract
The Hungarian Construction (Komlós et al. 1975) is used for getting a proof of asymptotic normality of S-Gini coefficient; this method is very interesting because it can be used to check asymptotic normality of other income inequality measures as Theil coefficient. Besides, explicit expressions of asymptotic means and variances are given for S-Gini and Theil estimators. Finally, to illustrate the performance of obtained results, we carry out a simulation study comparing the asymptotic and Smoothed Bootstrap approximations.
Key words: S-Gini index, Theil index, Hungarian construction, Kernel density estimation.
Se usa el Proceso Húngaro (Komlós et al. 1975) para derivar la normalidad asintótica del S-Gini; Este método es muy interesante ya que puede ser usado para demostrar la normalidad asintótica de otros coeficientes usados para medir la desigualdad de ingresos como el de Theil. Se consiguen expresiones explícitas para la media y la varianza del S-Gini y del coeficiente de Theil. Finalmente, se realiza un estudio de simulación, en el que se compara el rendimiento de la aproximación asintótica propuesta y del método Bootstrap Suavizado.
Palabras clave: índice S-Gini, índice de Theil, proceso húngaro, estimación kernel para la densidad.
Texto completo disponible en PDF
References
1. Billingsley, P. (1968), Convergence of Probability Measures, John Wiley and Sons, New York, United States. [ Links ]
2. Chakravarty, S. R. (1988), 'Extended Gini Indices of Inequality', International Economic Review 29, 147-156. [ Links ]
3. Crocetta, C. & Loperfido, N. (2005), 'The Exact Sampling Distribution of L-statistics', Metron LXIII(2), 293. [ Links ]
4. Donalson, D. & Weymark, J. A. (1980), 'A Single Parameter Generalization of the Gini Index and Inequality', Journal of Economic Theory 22, 67-86. [ Links ]
5. Donalson, D. & Weymark, J. A. (1983), 'Ethical Flexible Indices for Income Distributions in the Continuum', Journal of Economic Theory 29, 353-358. [ Links ]
6. Giles, E. (2004), 'Calculating a Standard Error for the Gini Coefficient: Some Further Results', Oxford Bulletin of Economics & Statistics 66, 425-428. [ Links ]
7. Gini, C. (1995), Variabilitá e mutabilita, 'Memorie di Metodologia Statistica', Rome, Italy. [ Links ] Reprinted (Libreria Eredi Virgilio Veschi, 1912).
8. González Manteiga, W., Prada Sánchez, J. M. & Romo, J. (1994), 'The Bootstrap: a Review', Computational Statistics 9, 165-205. [ Links ]
9. Hall, P., DiCiccio, J. T. & Romano, J. P. (1989), 'On Smoothing and the Bootstrap', Annals of Statistics 17(2), 692-704. [ Links ]
10. Komlós, J., Major, P. & Tusnády, G. (1975), 'An Approximation of Partial Sums of Independent RV-S, and the Sample DF.I.', Z. Wahrscheinlichkeitstheor. Verw. Geb. 32(1-2), 111-131. [ Links ]
11. Martínez-Camblor, P. (2005), 'Normalidad asintótica para los E-Gini', Revista de la Sociedad Argentina de Estadística 9, 35-42. [ Links ]
12. Martínez-Camblor, P. (2006), 'Tests no paramétricos basados en una medida de igualdad entre funciones de densidad', Servicio de Publicaciones de la Universidad de Oviedo, Oviedo, España. [ Links ]
13. Martínez-Camblor, P. (2007), 'Comparing S-Gini and E-Gini on Paired Samples', InterSTAT. * [ Links ]http://interstat.statjournals.net/YEAR/2007/abstracts/0705001.php
14. Metropolis, N. & Ulam, S. (1949), 'The Monte Carlo Method', Journal of the American Statistical Association 44, 335-345. [ Links ]
15. Nadaraya, E. A. (1964), 'Some New Estimates for Distribution Functions', Theory Prob. Applic. 9, 497-500. [ Links ]
16. R Development Core Team, (2006), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org [ Links ]
17. Theil, H. (1967), Economics and Information Theory, North-Holland, Amsterdam, Nederland. [ Links ]
18. Yitzhaki, S. (1983), 'On an Extension of the Gini Inequality Index', International Economic Review 24, 183-192. [ Links ]
19. Zitikis, R. (2003), 'Asymptotic Estimation of the E-Gini Index', Econometric Theory 19, 587-601. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{Martínez-Camblor07,
AUTHOR = {Pablo Martínez-Camblor}
TITLE = {{Central Limit Theorems for S-Gini and Theil Inequality Coefficients}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2007},
volume = {30},
number = {2},
pages = {287-300}
}