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 enrico.fabrizi@unicatt.it maria.ferrante@unibo.it carlo.trivisano@unibo.it

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

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 

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.

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.

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

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

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

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.

THE PRIORS SPECIFICATION

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

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