Poverty and inequality mapping based on a unit-level log-normal mixture model

Estimating poverty and inequality parameters for small sub-populations with adequate precision is often beyond the reach of ordinary survey-weighted methods because of small sample sizes. In small area estimation, survey data and auxiliary information are combined, in most cases using a model. In this paper, motivated by the analysis of EU-SILC data for Italy, we target the estimation of a selection of poverty and inequality indicators, that is mean, headcount ratio and quintile share ratio, adopting a Bayesian approach. We consider unit-level models specified on the log transformation of a skewed variable (equivalized income). We show how a finite mixture of log-normals provides a substantial improvement in the quality of fit with respect to a single log-normal model. Unfortunately, working with these distributions leads, for some estimands, to the non-existence of posterior moments whenever priors for the variance components are not carefully chosen, as our theoretical results show. To allow the use of moments in posterior summaries, we recommend generalized inverse Gaussian distributions as priors for variance components, guiding the choice of hyperparameters.