Download estimation of some inequality indexes

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  i1 N
 1,
N  xi
i1
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  i1
 1,
N
N  yi
i1
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
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Sampling, New York: Springer – Verlag
DEVILLE, J.-C. (1999) Variance Estimation for Complex Statistics and Estimators:
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DEVILLE, J.-C. AND SÄRNDAL, C.-E. (1992)., Calibration estimators in survey
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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.
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