Estimation of order statistics and income inequality measures:

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Estimation of order statistics and income
inequality measures

• Development and testing of new estimation tools
  for a large-scale production environment
• By
• Claes Andersson and Anders Holmberg

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• The estimation problem
• A short description of a general software for the
  estimation of functions of totals and order
  statistics in complex surveys, ETOS
• An empirical comparison of point and standard
  error estimators of the Gini index

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Let y be a variable of interest in a fix
population U of N units, let yk be the y-value of
unit k and let ? be a parameter of interest. A
sample s of size n is taken from U by the design
p() with first and second order inclusion
probabilities ?k and ?kl of units k and kl. The
parameter ? can be defined explicitly like the
total, or implicitly like the first quartile,
where I() is an indicator (0,1) function.
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• is the solution to the equation
• 
  for the first quantile.
• The solution
• is the Estimation Equation (EE) estimator of ?,
  where wk is 1/?k possibly adjusted for
  non-response and the impact of auxiliary
  information.

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The pth population quantile is defined as which
is estimated by with
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To estimate the variance of a Taylor
expansion of u() is used, The approximate
variance of is then The u() is estimated
by is readily obtained but how to find
?
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One suggestion is to calculate
by where are the upper and
lower limits of a 95 (say) confidence interval
around by using Woodruffs method.
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The Gini index in domain d is estimated
by with and The
-variable is defined by,
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ETOS a general software

• Totals, Quantiles, Gini index,
• Quantile totals
• Quantile proportion
• Rational functions of the parameters.
• Stratification, Two-stage designs, SRS, ?ps,
  Two-phase designs.
• Auxiliary variables, calibration.
• Different options for the treatment of
  non-response

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An empirical study of the Gini index estimation

• ETOS, model based estimator, bootstrap
• Survey of Households Finances
• Population of 6 7169 98316 699 households
  divided into two strata.
• Sampling fractions, 10 and 30, SRS
• 10 000 replications
• Variable, disposable income per consumption unit
• 4 domains the total population

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Measures of performance
Coverage rate and upper and lower tail error rates
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Conclusions

• Approximately unbiased estimators.
• The variance estimators are approximately
  unbiased and performs similarly.
• The EE estimator used in ETOS gives a slightly
  better coverage rate of the 95 CI.
• The sampling distribution of is skew and
  deviates from the Normal distribution