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.

中文翻译：

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基于单位级对数正态混合模型的贫困和不平等映射

由于样本量小，普通调查加权方法往往无法以足够的精度估算小部分人口的贫困和不平等参数。在小区域估计中，调查数据和辅助信息相结合，在大多数情况下使用模型。在本文中，受意大利 EU-SILC 数据分析的启发，我们采用贝叶斯方法来估计一系列贫困和不平等指标，即平均人口比率和五分位数份额比率。我们考虑在偏态变量（等值收入）的对数变换上指定的单元级模型。我们展示了对数正态分布的有限混合如何相对于单个对数正态分布模型显着提高拟合质量。不幸的是，对于某些估计，使用这些分布导致，正如我们的理论结果所示，只要没有仔细选择方差分量的先验，就会导致不存在后验矩。为了允许在后验摘要中使用矩，我们建议将广义逆高斯分布作为方差分量的先验，指导超参数的选择。