General Restriction Estimator in Small Area Estimation

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General Restriction Estimator in Small Area
Estimation

• Kaja Sõstra
• Statistics Estonia
• European Conference on Quality in Survey
  Statistics
• 24-26 April 2006

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Small area estimation problem

• National sample surveys are usually designed for
  estimating at national level and in large groups
• Sample size in small area (geographical or
  subgroup) is often too small for reliable direct
  estimates
• Several methods has been designed for borrowing
  strength from neighbouring regions

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General regression estimator (GREG)

• The generalised regression estimator (GREG) for
  an area is obtained by adjusting the direct
  estimator using the standard linear model.

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General regression estimator (GREG)

• is the vector of true totals of p covariates
  in the area d
• is the least squares regression estimate
  assuming a standard linear model
• with independent errors

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SAE and population total estimator

• SAE perform differently depending on the size of
  area
• For better results it is appropriate to use
  different estimates for smaller areas and large
  sub-populations
• Obtained estimates do not satisfy the criteria
  that the sum of estimated small area totals is
  equal to estimated population totals
• One solution of the problem is general
  restriction estimator

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General restriction (GR) estimator

• Consider a k-vector of unbiased estimators
• The nonsingular covariance matrix is
• The parameters have to obey the set of m linear
  restrictions
• where R is m x k matrix of rank m

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GR estimator (2)

• The general restriction estimator that satisfies
  restrictions above is (Knottnerus, 2003)

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GR estimator (example)

• Consider two different samples from the same
  population.
• Estimates for population totals Y and Z are
  obtained from sample 1 and for population totals
  U and Z from sample 2.
• Parameters
• Restriction

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GR estimator for SAE (1)

• Vector of estimated small area totals and vector
  R
• Restriction small area totals have to sum up to
  population total Y

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GR estimator for SAE (2)

• For calculating restriction estimators the
  covariance matrix of the parameter vector is
  needed. Assuming the independence of regions the
  covariance matrix takes a form

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GR estimator of small area total

• Assuming that population total Y is known from
  external sources or estimated from larger survey
  with high precision the GR estimator for small
  area d takes a form

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Variance of GR estimator

• Variance estimator of GR estimator of small area
  d
• Variance of GR estimator is decreased by

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Simulation study

• Population was generated for simulation study
  with target variable Y and auxiliary variable X
• Population size N 10,000
• Number of small areas D 3
• Y and X were normally distributed with
  correlation 0.85
• 1000 samples were drawn from population with
  sample size n 100
• Small area totals were estimated using GREG
  estimator.
• GR estimators were calculated using known
  population total (GR1) and population total GREG
  estimator (GR2)

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Performance of estimators

• Measures for comparing the performance of
  estimators
• 1) Absolute relative bias (ARB)
• 2) Relative root mean square error (RRMSE)

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Performance criteria by small area
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Conclusions

• GR estimator is a possibility for calibration the
  small area estimators to meet certain conditions.
• Simulation study showed that GR estimator
  performed slightly better than GREG estimator