Post-Stratified Synthetic Estimator (PS-SYN)

Overview

• Focus on post-stratified synthetic estimator.
• Rarely used in small area estimation due to unrealistic assumptions.

Construction of the Estimator

• Good direct estimates known for cross-classifications (e.g., gender, age, education).
• If population sizes or totals of auxiliary variables are known, a synthetic estimator can be constructed.

Key Definitions

• Cross-classifications: Translated into qualitative variables related to target variables.
• Population (U) divided into j groups with known sizes (n1, n2, ... nj).
• Area (D) divided into J groups (post-strata) with known means (green colored means for each stratum).

Assumptions

• Homogeneity assumption: Individuals in each stratum behave homogeneously, regardless of area.
  • For each domain, mean of stratum J equals mean of the whole stratum J.
  • Example: Average disposable income in area D.

Estimation Process

• Estimate area mean for domain D by estimating post-strata means.
• Strata means estimated using AGEK estimators.
• Small variance expected due to large sample sizes in strata.

Issues with Assumption

• Homogeneity assumption may not hold true (e.g., women’s income varies by area).
• Results in potential bias, affecting mean square error (MSE).

Application to Poverty Metrics

• Can be applied to Foster-Grier-Torbeke (FGT) indicators, a family of poverty metrics.
  • FjAlphaD: FGT indicator for the domain.
  • Headcount Index: Proportion of population that is poor (may not reflect severity).
  • Poverty Gap Index: Measures extent below poverty line (may not reflect true situation).

Properties of PS-SYN Estimator

• Target indicators: mean or total of target variable.
• Data requirements:
  • Design weights for all sample individuals.
  • Population size of area ND, by strata classification.
  • Qualitative variable(s) related to target variables.

Pros and Cons

Pros

• Variance reduction compared to direct estimator when strata have adequate observations.

Cons

• Homogeneity assumption often unrealistic, especially with gender stratification.
• Bias can negate variance reduction benefits.
• Difficulty in estimating stable MSE combining variance and bias contributions.

Conclusion

• PS-SYN estimator has potential but relies heavily on the validity of assumptions.
• Caution is advised when applying to heterogeneous populations.