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ESTIMATION OF SOME INEQUALITY INDEXES Inga Masiulaitytė Statistics Lithuania, Vilnius University e-mail: [email protected] Inequality indexes are important indicators in analysing household welfare. Inequality indexes such as the Gini coefficient, concentration index of expenditure and elasticity index are often met in official statistics. These indexes have a complicated construction. We try to analyze the behaviour of each index, and different methods for their variance estimation. The Jackknife method, Rescaling bootstrap and the Deville method are selected to variance estimation. The Gini coefficient and the concentration index of expenditure Suppose we have a finite population U consisting of N individuals: U {1,..., N } . Let the probability sample s of size n be taken from the population U according to any sampling design. Denote by k P( s : k s) the inclusion probability into the sample s, by d k 1 / k , k U , sample design weights. Let us investigate variables of interest X (income) and Y (expenditures) with the values in the population x1 , x2 ,..., xN and y1 , y2 ,..., y N . Let us investigate the indexes that characterize the inequality of income and expenditures of the population: the Gini coefficient N ( 2r (i ) 1) xi Gx i1 N 1, N xi i1 N r (i ) I x x k 1 k i 1, I x x k i 0, xk xi ; , otherwise, and the concentration index of expenditure N ( 2q (i ) 1) yi C y i1 1, N N yi i1 N q (i ) I y y k 1 k i 1, I y y k i 0, y k yi ; otherwise. Here r(i) is the rank of the individual i in a sequence of individuals arranged in an ascending order by income, q(i) is the rank of the individual i in a sequence of individuals arranged in the ascending order by expenditure. Elasticity index The deviation of one variable from proportionality with respect to another variable is measured by elasticity. The elasticity index is defined by: E Cy , Gx where Cy is the concentration index of total expenditure (or expenditure on a fixed item), Gx is the Gini coefficient of income (or total expenditure). Index of the deviation of elasticity from a unit The index of the elasticity deviation from a unit is defined (Podder, 1995) as follows: I E 1 C y Gx , 2 where Cy is the concentration index of total expenditure (or expenditure on item), Gx is the Gini coefficient of income (or total expenditure). Simulation results and conclusion will be presented later. Some calculations are still in progress. References SÄRNDAL, C.-E., SWENSSON, B., WRETMAN J. (1992), Model Assisted Survey Sampling, New York: Springer – Verlag DEVILLE, J.-C. (1999) Variance Estimation for Complex Statistics and Estimators: Linearization and Residual Technique, Survey Methodology, Vol. 25, No.2, pp. 193-203. DEVILLE, J.-C. AND SÄRNDAL, C.-E. (1992)., Calibration estimators in survey sampling, Journal of the American Statistical Association, Vol. 87, pp. 376-382. PODDER, N. (1995), On the relationship between the Gini coefficient and income elasticity. Sankhya. The Indian Journal of Statistics, Vol. 57, Series B, Pt. 3, pp.428–432. KHANDKER, S. (2004), Poverty Manual, World Bank. DELL, F., D’HAULTFEUILLE, X., FÉVRIER, P. AND MASSÉ, E. (2004), Linearization method of variance estimation, implementation and application to the French “Tax Income Survey”. Working paper, INSEE, Manuscript. RAO, J. N. K. AND WU, C. F. J. (1988), Resampling Inference With Complex Survey Data. Journal of the American Statistical Association, Vol. 83, No. 401, pp. 231-241.