1. Enrico Fabrizi°, Maria Rosaria Ferrante* , Carlo Trivisano*
Conference of European Statistics Stakeholders
Budapest, 20–21 October 2016
SMALL AREA ESTIMATION OF THE RELATIVE MEDIAN POVERTY GAP
Enrico Fabrizi°, Maria Rosaria Ferrante* , Carlo Trivisano*
*Department of Statistical Sciences - University of Bologna
°DISES, Universita Cattolica, Piacenza

2. AIM OF THE PAPER
Estimation of the Relative median at-risk-of-poverty gap (GAP) in small areas
i=1,…,m areas
NPT national poverty threshold (60% of the median)
individuals below the threshold, j=1,…,n units
median income for individuals below the threshold
The GAP
measures the depth of poverty: “how much the poor people are poor?”
complements the most frequently used At-risk-of-poverty rate

3. PROBLEMS TO DEAL WITH
Challenges in the estimation of the GAP in Small Areas:
the area-specific sample sizes is not large enough to obtain reliable estimates form survey-weighted estimators (direct estimators)
the GAP suffers particularly of this problem as it is defined only on the poor sub-population income distribution: the direct estimation is based only on the income of the individuals below the NPT (a minority of the sample units)
the direct estimator of the GAP is not only highly variable in small samples, but it is also biased, as estimators of quantiles usually are 

4. SMALL AREA ESTIMATION (SAE) MODELS
Small area estimation models complement the insufficient information at small area level with auxiliary information known for the population.
Models can be specified at either the area or the individual level
We consider area-level models requiring:
area level direct estimator with their variability
but design-based estimator of the GAP is biased in small samples
area-level auxiliary information with good predictive power
but for the GAP they are difficult to find
For other summaries of the income distribution (mean, rates or Gini coefficient) we dispose of unbiased or approximately unbiased direct estimators, for which it is more likely to find effective predictors.

5. PROPOSAL
We assume a parametric model for the distribution of income, with different parameters in each area. Here we evaluate the LogNormal distribution but other distributional assumptions are under test (GB2)
We select a set of population summaries other than GAP for which SAE models can be estimated: here we focus on the at-risk-of-poverty rate, the log income average and the Gini concentration index
Under the distributional assumption, the GAP is a known function of the parameters estimated through SAE model
The approach to estimation is Bayesian We approximate posterior distributions of these summary statistics (given the data and the auxiliary variables) by means of MCMC algorithms, and we generate a Markov Chain for the GAP.

6. THE DATA 2008-2013 Italian EU-SILC sample survey
auxiliary information: publicly available archives of the Ministry of Economics and Finance and of Italian National Statistical Institute at the municipal level
target small areas: administrative provinces playing an important role in the implementation of many social policies related to the contrast of poverty and social exclusion
number of small areas: 110
area specific sample sizes range from 6 to 2647
number of poor households in small areas: min=0, max=217, median=22
number of poor persons in small areas: min=0, max=608, median=49

7. THE SMALL AREA MODEL (1)
For the At-Risk-of-Poverty rate and for the Gini index
i=1,…,m areas, t=1,…,T
Sampling model: Beta model
Linking model: Logit model
pit direct estimator of
direct estimator of

8. THE SMALL AREA MODEL (2) For the log income average
Sampling model: Normal model
Linking model: Normal model
direct estimator for

9. RELATING PARAMETERS THROUGH THE DISTRIBUTIONAL ASSUMPTION
Distributional assumption for income Z:
Parameter to be stimated (GAP): with
Assuming known the NPT:
Given the solution of leads to
Posterior for is generated from the posterior distributions of
Posterior mean is negatively biased and then benchmarked.

10. THE PRIORS SPECIFICATION

11. THE RESULTS
Fig. - Difference between design and model based estimates against sample size
The differences approach 0 as the domain-specific sample sizes grow large

12. THANK YOU
FOR YOUR ATTENTION!